{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", " Asymptotics of 2-SAT\n", "\n", "\n", "This is a supplementary file for the paper\n", " \n", " \"Asymptotics for graphically divergent series: dense digraphs and 2-SAT formulae\"\n", " by Sergey Dovgal and Khaydar Nurligareev.\n", " \n", "Here, you can find the code for obtaining asymptotic coefficients from Sections A.11-A.13, Tables 23-29" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", " Preliminary section\n", "" ] }, { "cell_type": "code", "execution_count": 253, "metadata": {}, "outputs": [], "source": [ "N = 20 # Our accuracy\n", "disp = 10 # How many terms we want to display\n", "P. = PolynomialRing(QQ)\n", "PP. = PolynomialRing(P)\n", "PQ. = PolynomialRing(PP)\n", "PR. = PolynomialRing(PQ)\n", "R. = PowerSeriesRing(PR,N)" ] }, { "cell_type": "code", "execution_count": 254, "metadata": {}, "outputs": [], "source": [ "# Exponential Hadamard product of two series\n", "# order is the accuracy of this operation\n", "def exp_had_prod(f, g, order):\n", " return sum(f[n] * g[n] * factorial(n) * z^n for n in range(order))" ] }, { "cell_type": "code", "execution_count": 255, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1, 1, 1/a^4, 1/a^12, 1/a^24, 1/a^40, 1/a^60, 1/a^84, 1/a^112, 1/a^144]" ] }, "execution_count": 255, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Transformation function hat(set)\n", "hatset = sum(((1 / a^(2*i*(i-1))) / i.factorial()) * z^i for i in srange(N))\n", "HATSET = [hatset[i] * i.factorial() for i in srange(disp)]\n", "HATSET" ] }, { "cell_type": "code", "execution_count": 256, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1, 1, 1/a^8, 1/a^24, 1/a^48, 1/a^80, 1/a^120, 1/a^168, 1/a^224, 1/a^288]" ] }, "execution_count": 256, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Transformation function ddot(set)(2z)\n", "ddotset = sum(((1 / a^(4*i*(i-1))) / i.factorial()) * z^i for i in srange(N))\n", "DDOTSET = [ddotset[i] * i.factorial() for i in srange(disp)]\n", "DDOTSET" ] }, { "cell_type": "code", "execution_count": 257, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1, 1, a^2, a^6, a^12, a^20, a^30, a^42, a^56, a^72]" ] }, "execution_count": 257, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Transformation function Phi_{a^2}^{1,2}\n", "PhiTwoOneTwo = sum((a^(i*(i-1)) / i.factorial()) * z^i for i in srange(N))\n", "PTOT = [PhiTwoOneTwo[i] * i.factorial() for i in srange(disp)]\n", "PTOT" ] }, { "cell_type": "code", "execution_count": 258, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1, 1, 1/a^2, 1/a^6, 1/a^12, 1/a^20, 1/a^30, 1/a^42, 1/a^56, 1/a^72]" ] }, "execution_count": 258, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Transformation function Phi_{a^2}^{2,1}\n", "PhiTwoTwoOne = sum((1 / a^(i*(i-1)) / i.factorial()) * z^i for i in srange(N))\n", "PTTO = [PhiTwoTwoOne[i] * i.factorial() for i in srange(disp)]\n", "PTTO" ] }, { "cell_type": "code", "execution_count": 259, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1, 1, a, a^3, a^6, a^10, a^15, a^21, a^28, a^36]" ] }, "execution_count": 259, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Transformation function Phi_{a^2}^{2,4}\n", "PhiTwoTwoFour = sum((a^(i*(i-1)/2) / i.factorial()) * z^i for i in srange(N))\n", "PTTF = [PhiTwoTwoFour[i] * i.factorial() for i in srange(disp)]\n", "PTTF" ] }, { "cell_type": "code", "execution_count": 260, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1, 1, 1/a, 1/a^3, 1/a^6, 1/a^10, 1/a^15, 1/a^21, 1/a^28, 1/a^36]" ] }, "execution_count": 260, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Transformation function Phi_{a^2}^{4,2}\n", "PhiTwoFourTwo = sum((1 / (a^(i*(i-1)/2)) / i.factorial()) * z^i for i in srange(N))\n", "PTFT = [PhiTwoFourTwo[i] * i.factorial() for i in srange(disp)]\n", "PTFT" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", " Defining supplementary EGFs\n", "" ] }, { "cell_type": "code", "execution_count": 261, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1, 1, a^4, a^12, a^24, a^40, a^60, a^84, a^112, a^144]" ] }, "execution_count": 261, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# EGF of labeled graphs/tournaments or GGF of labeled digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "g = sum((a^(2*i*(i-1)) / i.factorial()) * z^i for i in srange(N))\n", "G = [g[i] * i.factorial() for i in srange(disp)]\n", "G" ] }, { "cell_type": "code", "execution_count": 262, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " 1,\n", " a^4 - 2,\n", " a^12 - 6*a^4 + 6,\n", " a^24 - 8*a^12 - 6*a^8 + 36*a^4 - 24,\n", " a^40 - 10*a^24 - 20*a^16 + 60*a^12 + 90*a^8 - 240*a^4 + 120,\n", " a^60 - 12*a^40 - 30*a^28 + 70*a^24 + 360*a^16 - 390*a^12 - 1080*a^8 + 1800*a^4 - 720,\n", " a^84 - 14*a^60 - 42*a^44 + 126*a^40 - 70*a^36 + 630*a^28 - 420*a^24 + 630*a^20 - 5040*a^16 + 1680*a^12 + 12600*a^8 - 15120*a^4 + 5040,\n", " a^112 - 16*a^84 - 56*a^64 + 168*a^60 - 112*a^52 - 70*a^48 + 1008*a^44 - 1344*a^40 + 1680*a^36 + 1260*a^32 - 8400*a^28 + 1680*a^24 - 20160*a^20 + 64680*a^16 + 10080*a^12 - 151200*a^8 + 141120*a^4 - 40320,\n", " a^144 - 18*a^112 - 72*a^88 + 216*a^84 - 168*a^72 + 1260*a^64 - 2016*a^60 + 3024*a^52 + 4158*a^48 - 18144*a^44 + 22680*a^40 - 28560*a^36 - 45360*a^32 + 90720*a^28 - 20160*a^24 + 453600*a^20 - 793800*a^16 - 483840*a^12 + 1905120*a^8 - 1451520*a^4 + 362880]" ] }, "execution_count": 262, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# EGF of irreducible labeled tournaments (we take a parameter a = sqrt{sqrt{2}})\n", "it = 1 - 1/g\n", "IT = [it[i] * i.factorial() for i in srange(disp)]\n", "IT" ] }, { "cell_type": "code", "execution_count": 263, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0, 1, 0, 2, 24, 544, 22320, 1677488, 236522496, 64026088576]" ] }, "execution_count": 263, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Numbers of irreducible labeled tournaments\n", "ITsub = [it[i].subs(a=sqrt(sqrt(2))) * i.factorial() for i in srange(disp)]\n", "ITsub" ] }, { "cell_type": "code", "execution_count": 264, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1, 1, a^8, a^24, a^48, a^80, a^120, a^168, a^224, a^288]" ] }, "execution_count": 264, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# EGF of labeled digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "d = sum((a^(4*i*(i-1)) / i.factorial()) * z^i for i in srange(N))\n", "D = [d[i] * i.factorial() for i in srange(disp)]\n", "D" ] }, { "cell_type": "code", "execution_count": 265, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " 1,\n", " a^8 - 2*a^4 + 1,\n", " a^24 - 6*a^16 + 6*a^12 + 3*a^8 - 6*a^4 + 2,\n", " a^48 - 8*a^36 - 6*a^32 + 36*a^28 - 20*a^24 - 21*a^16 + 12*a^12 + 24*a^8 - 24*a^4 + 6,\n", " a^80 - 10*a^64 - 20*a^56 + 60*a^52 + 95*a^48 - 240*a^44 + 120*a^40 - 40*a^36 - 20*a^32 + 160*a^28 - 160*a^24 + 180*a^20 - 210*a^16 + 180*a^8 - 120*a^4 + 24,\n", " a^120 - 12*a^100 - 30*a^88 + 70*a^84 + 6*a^80 + 360*a^76 - 390*a^72 - 1080*a^68 + 1740*a^64 - 720*a^60 - 105*a^56 + 330*a^52 + 580*a^48 - 1560*a^44 + 750*a^40 + 600*a^36 - 1140*a^32 + 840*a^28 - 930*a^24 + 1980*a^20 - 1530*a^16 - 600*a^12 + 1440*a^8 - 720*a^4 + 120,\n", " a^168 - 14*a^144 - 42*a^128 + 126*a^124 - 63*a^120 + 630*a^112 - 420*a^108 + 630*a^104 - 5124*a^100 + 1680*a^96 + 12600*a^92 - 15309*a^88 + 5488*a^84 + 42*a^80 + 2520*a^76 - 2905*a^72 - 7140*a^68 + 11550*a^64 - 3010*a^60 - 1470*a^56 - 3780*a^52 + 15470*a^48 - 25620*a^44 + 18270*a^40 + 4200*a^36 - 15540*a^32 + 11340*a^28 - 16170*a^24 + 23940*a^20 - 10080*a^16 - 10080*a^12 + 12600*a^8 - 5040*a^4 + 720,\n", " a^224 - 16*a^196 - 56*a^176 + 168*a^172 + 8*a^168 - 112*a^164 - 70*a^160 + 1008*a^156 - 1344*a^152 + 1680*a^148 + 1148*a^144 - 8400*a^140 + 1680*a^136 - 20160*a^132 + 64372*a^128 + 11032*a^124 - 151704*a^120 + 141120*a^116 - 35280*a^112 - 3696*a^108 + 5768*a^104 - 40992*a^100 + 12299*a^96 + 104776*a^92 - 126784*a^88 + 53088*a^84 - 28924*a^80 + 14280*a^76 + 98560*a^72 - 211680*a^68 + 105840*a^64 + 99680*a^60 - 80920*a^56 - 103600*a^52 + 239400*a^48 - 290640*a^44 + 178080*a^40 + 47040*a^36 - 127050*a^32 + 139440*a^28 - 270480*a^24 + 282240*a^20 - 45360*a^16 - 141120*a^12 + 120960*a^8 - 40320*a^4 + 5040,\n", " a^288 - 18*a^256 - 72*a^232 + 216*a^228 + 9*a^224 - 168*a^216 + 1260*a^208 - 2016*a^204 + 2880*a^196 + 4158*a^192 - 18144*a^188 + 22680*a^184 - 28560*a^180 - 45828*a^176 + 92160*a^172 - 20088*a^168 + 452592*a^164 - 794430*a^160 - 474768*a^156 + 1892520*a^152 - 1435392*a^148 + 373296*a^144 - 75600*a^140 + 13104*a^136 - 173376*a^132 + 568134*a^128 + 102816*a^124 - 1342656*a^120 + 1214640*a^116 - 275184*a^112 - 252672*a^108 + 491400*a^104 - 38808*a^100 - 1328922*a^96 + 1918224*a^92 - 1010520*a^88 + 620928*a^84 - 1473696*a^80 + 1658160*a^76 + 108920*a^72 - 1895040*a^68 + 525420*a^64 + 1975680*a^60 - 695520*a^56 - 3296160*a^52 + 5720400*a^48 - 4959360*a^44 + 1648080*a^40 + 1043280*a^36 - 1459080*a^32 + 2600640*a^28 - 4505760*a^24 + 3265920*a^20 + 272160*a^16 - 1935360*a^12 + 1270080*a^8 - 362880*a^4 + 40320]" ] }, "execution_count": 265, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# EGF of strongly connected labeled digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "scd = -log(exp_had_prod(g,1/g,N))\n", "SCD = [scd[i] * i.factorial() for i in srange(disp)]\n", "SCD" ] }, { "cell_type": "code", "execution_count": 266, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " t,\n", " ((2*a^4 - 1)/a^4)*t^2 + ((a^8 - 2*a^4 + 1)/a^4)*t,\n", " ((6*a^12 - 6*a^8 + 1)/a^12)*t^3 + ((6*a^16 - 12*a^12 + 3*a^8 + 6*a^4 - 3)/a^12)*t^2 + ((a^24 - 6*a^16 + 6*a^12 + 3*a^8 - 6*a^4 + 2)/a^12)*t,\n", " ((24*a^24 - 36*a^20 + 6*a^16 + 8*a^12 - 1)/a^24)*t^4 + ((36*a^28 - 84*a^24 + 36*a^20 + 36*a^16 - 24*a^12 + 6*a^8 - 12*a^4 + 6)/a^24)*t^3 + ((8*a^36 + 6*a^32 - 72*a^28 + 80*a^24 - 21*a^16 + 4*a^12 - 30*a^8 + 36*a^4 - 11)/a^24)*t^2 + ((a^48 - 8*a^36 - 6*a^32 + 36*a^28 - 20*a^24 - 21*a^16 + 12*a^12 + 24*a^8 - 24*a^4 + 6)/a^24)*t,\n", " ((120*a^40 - 240*a^36 + 90*a^32 + 60*a^28 - 20*a^24 - 10*a^16 + 1)/a^40)*t^5 + ((240*a^44 - 660*a^40 + 420*a^36 + 260*a^32 - 340*a^28 + 140*a^24 - 120*a^20 + 60*a^16 - 10*a^8 + 20*a^4 - 10)/a^40)*t^4 + ((60*a^52 + 70*a^48 - 720*a^44 + 920*a^40 - 60*a^36 - 480*a^32 + 360*a^28 - 390*a^24 + 360*a^20 - 155*a^16 + 120*a^8 - 120*a^4 + 35)/a^40)*t^3 + ((10*a^64 + 20*a^56 - 120*a^52 - 165*a^48 + 720*a^44 - 500*a^40 - 80*a^36 + 150*a^32 - 240*a^28 + 430*a^24 - 420*a^20 + 315*a^16 - 290*a^8 + 220*a^4 - 50)/a^40)*t^2 + ((a^80 - 10*a^64 - 20*a^56 + 60*a^52 + 95*a^48 - 240*a^44 + 120*a^40 - 40*a^36 - 20*a^32 + 160*a^28 - 160*a^24 + 180*a^20 - 210*a^16 + 180*a^8 - 120*a^4 + 24)/a^40)*t,\n", " ((720*a^60 - 1800*a^56 + 1080*a^52 + 390*a^48 - 360*a^44 - 70*a^36 + 30*a^32 + 12*a^20 - 1)/a^60)*t^6 + ((1800*a^64 - 5760*a^60 + 4950*a^56 + 1620*a^52 - 4050*a^48 + 1860*a^44 - 1050*a^40 + 840*a^36 - 210*a^32 - 120*a^28 + 240*a^24 - 120*a^20 + 15*a^8 - 30*a^4 + 15)/a^60)*t^5 + ((480*a^72 + 720*a^68 - 7470*a^64 + 11200*a^60 - 2220*a^56 - 6450*a^52 + 5160*a^48 - 3540*a^44 + 4260*a^40 - 3070*a^36 + 510*a^32 + 1440*a^28 - 1460*a^24 + 420*a^20 + 75*a^16 + 60*a^12 - 330*a^8 + 300*a^4 - 85)/a^60)*t^4 + ((90*a^84 - 30*a^80 + 360*a^76 - 1350*a^72 - 2820*a^68 + 11370*a^64 - 9240*a^60 - 1380*a^56 + 4110*a^52 - 2535*a^48 + 4800*a^44 - 7230*a^40 + 5700*a^36 - 540*a^32 - 3060*a^28 + 2055*a^24 + 390*a^20 - 990*a^16 - 480*a^12 + 1605*a^8 - 1050*a^4 + 225)/a^60)*t^3 + ((12*a^100 + 30*a^88 - 160*a^84 + 24*a^80 - 720*a^76 + 1260*a^72 + 3180*a^68 - 7440*a^64 + 3800*a^60 + 555*a^56 - 690*a^52 + 455*a^48 - 1200*a^44 + 3270*a^40 - 4000*a^36 + 1350*a^32 + 900*a^28 + 95*a^24 - 2682*a^20 + 2445*a^16 + 1020*a^12 - 2730*a^8 + 1500*a^4 - 274)/a^60)*t^2 + ((a^120 - 12*a^100 - 30*a^88 + 70*a^84 + 6*a^80 + 360*a^76 - 390*a^72 - 1080*a^68 + 1740*a^64 - 720*a^60 - 105*a^56 + 330*a^52 + 580*a^48 - 1560*a^44 + 750*a^40 + 600*a^36 - 1140*a^32 + 840*a^28 - 930*a^24 + 1980*a^20 - 1530*a^16 - 600*a^12 + 1440*a^8 - 720*a^4 + 120)/a^60)*t,\n", " ((5040*a^84 - 15120*a^80 + 12600*a^76 + 1680*a^72 - 5040*a^68 + 630*a^64 - 420*a^60 + 630*a^56 - 70*a^48 + 126*a^44 - 42*a^40 - 14*a^24 + 1)/a^84)*t^7 + ((15120*a^88 - 55440*a^84 + 60480*a^80 + 5040*a^76 - 48510*a^72 + 28980*a^68 - 12600*a^64 + 11340*a^60 - 3780*a^56 - 2520*a^52 + 3612*a^48 - 2184*a^44 + 462*a^40 + 210*a^32 - 420*a^28 + 210*a^24 - 21*a^8 + 42*a^4 - 21)/a^84)*t^6 + ((4200*a^96 + 7560*a^92 - 82530*a^88 + 143640*a^84 - 49770*a^80 - 85680*a^76 + 89950*a^72 - 52920*a^68 + 57540*a^64 - 43470*a^60 + 2310*a^56 + 24780*a^52 - 25130*a^48 + 11130*a^44 - 840*a^40 + 840*a^36 - 4620*a^32 + 4200*a^28 - 1155*a^24 - 105*a^16 - 210*a^12 + 735*a^8 - 630*a^4 + 175)/a^84)*t^5 + ((840*a^108 - 630*a^104 + 5040*a^100 - 15470*a^96 - 40950*a^92 + 169680*a^88 - 159740*a^84 - 18060*a^80 + 102900*a^76 - 77210*a^72 + 100380*a^68 - 127050*a^64 + 74130*a^60 + 13860*a^56 - 63840*a^52 + 51835*a^48 - 9660*a^44 - 10290*a^40 - 6440*a^36 + 22470*a^32 - 15540*a^28 + 4795*a^24 - 3150*a^20 + 2310*a^16 + 2940*a^12 - 5985*a^8 + 3570*a^4 - 735)/a^84)*t^4 + ((126*a^124 - 42*a^120 + 630*a^112 - 2100*a^108 + 1386*a^104 - 14700*a^100 + 19040*a^96 + 70770*a^92 - 164850*a^88 + 91980*a^84 + 26061*a^80 - 44100*a^76 + 58800*a^72 - 108780*a^68 + 116970*a^64 - 64260*a^60 - 10815*a^56 + 56910*a^52 - 34580*a^48 - 25956*a^44 + 33957*a^40 + 18060*a^36 - 45990*a^32 + 29190*a^28 - 17101*a^24 + 21420*a^20 - 11235*a^16 - 12390*a^12 + 19425*a^8 - 9450*a^4 + 1624)/a^84)*t^3 + ((14*a^144 + 42*a^128 - 252*a^124 + 105*a^120 - 1260*a^112 + 1680*a^108 - 1386*a^104 + 14784*a^100 - 9450*a^96 - 49980*a^92 + 77889*a^88 - 30968*a^84 - 3633*a^80 + 6720*a^76 - 21805*a^72 + 44520*a^68 - 47040*a^64 + 25690*a^60 - 735*a^56 - 11550*a^52 - 11137*a^48 + 52164*a^44 - 41517*a^40 - 16660*a^36 + 43470*a^32 - 28770*a^28 + 29435*a^24 - 42210*a^20 + 19110*a^16 + 19740*a^12 - 26754*a^8 + 11508*a^4 - 1764)/a^84)*t^2 + ((a^168 - 14*a^144 - 42*a^128 + 126*a^124 - 63*a^120 + 630*a^112 - 420*a^108 + 630*a^104 - 5124*a^100 + 1680*a^96 + 12600*a^92 - 15309*a^88 + 5488*a^84 + 42*a^80 + 2520*a^76 - 2905*a^72 - 7140*a^68 + 11550*a^64 - 3010*a^60 - 1470*a^56 - 3780*a^52 + 15470*a^48 - 25620*a^44 + 18270*a^40 + 4200*a^36 - 15540*a^32 + 11340*a^28 - 16170*a^24 + 23940*a^20 - 10080*a^16 - 10080*a^12 + 12600*a^8 - 5040*a^4 + 720)/a^84)*t,\n", " ((40320*a^112 - 141120*a^108 + 151200*a^104 - 10080*a^100 - 64680*a^96 + 20160*a^92 - 1680*a^88 + 8400*a^84 - 1260*a^80 - 1680*a^76 + 1344*a^72 - 1008*a^68 + 70*a^64 + 112*a^60 - 168*a^52 + 56*a^48 + 16*a^28 - 1)/a^112)*t^8 + ((141120*a^116 - 584640*a^112 + 776160*a^108 - 104160*a^104 - 588000*a^100 + 470400*a^96 - 179760*a^92 + 137760*a^88 - 63840*a^84 - 33600*a^80 + 53088*a^76 - 36456*a^72 + 11312*a^68 + 2072*a^64 + 1064*a^60 - 5936*a^56 + 4312*a^52 - 896*a^48 - 336*a^36 + 672*a^32 - 336*a^28 + 28*a^8 - 56*a^4 + 28)/a^112)*t^7 + ((40320*a^124 + 84000*a^120 - 977760*a^116 + 1950480*a^112 - 991200*a^108 - 1098720*a^104 + 1559040*a^100 - 871920*a^96 + 720720*a^92 - 594160*a^88 + 49952*a^84 + 362460*a^80 - 408912*a^76 + 215992*a^72 - 13944*a^68 - 16772*a^64 - 54656*a^60 + 73920*a^56 - 33880*a^52 + 5600*a^48 - 1680*a^44 - 3360*a^40 + 11760*a^36 - 10080*a^32 + 2800*a^28 - 56*a^24 + 126*a^16 + 504*a^12 - 1428*a^8 + 1176*a^4 - 322)/a^112)*t^6 + ((8400*a^136 - 10080*a^132 + 68460*a^128 - 189840*a^124 - 584640*a^120 + 2578800*a^116 - 2816660*a^112 - 54320*a^108 + 2101680*a^104 - 1723400*a^100 + 1595860*a^96 - 1904000*a^92 + 1138760*a^88 + 239120*a^84 - 1127140*a^80 + 1040760*a^76 - 248920*a^72 - 273280*a^68 + 47880*a^64 + 334040*a^60 - 336560*a^56 + 149240*a^52 - 67690*a^48 + 36960*a^44 + 47040*a^40 - 96320*a^36 + 57260*a^32 - 10360*a^28 - 3780*a^24 + 7560*a^20 - 3570*a^16 - 10920*a^12 + 17500*a^8 - 9800*a^4 + 1960)/a^112)*t^5 + 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35280*a^112 - 3696*a^108 + 5768*a^104 - 40992*a^100 + 12299*a^96 + 104776*a^92 - 126784*a^88 + 53088*a^84 - 28924*a^80 + 14280*a^76 + 98560*a^72 - 211680*a^68 + 105840*a^64 + 99680*a^60 - 80920*a^56 - 103600*a^52 + 239400*a^48 - 290640*a^44 + 178080*a^40 + 47040*a^36 - 127050*a^32 + 139440*a^28 - 270480*a^24 + 282240*a^20 - 45360*a^16 - 141120*a^12 + 120960*a^8 - 40320*a^4 + 5040)/a^112)*t,\n", " ((362880*a^144 - 1451520*a^140 + 1905120*a^136 - 483840*a^132 - 793800*a^128 + 453600*a^124 - 20160*a^120 + 90720*a^116 - 45360*a^112 - 28560*a^108 + 22680*a^104 - 18144*a^100 + 4158*a^96 + 3024*a^92 - 2016*a^84 + 1260*a^80 - 168*a^72 + 216*a^60 - 72*a^56 - 18*a^32 + 1)/a^144)*t^9 + ((1451520*a^148 - 6713280*a^144 + 10523520*a^140 - 3538080*a^136 - 7166880*a^132 + 7832160*a^128 - 3144960*a^124 + 1753920*a^120 - 1103760*a^116 - 378000*a^112 + 910224*a^108 - 675864*a^104 + 260064*a^100 + 28728*a^96 - 9072*a^92 - 80640*a^88 + 70560*a^84 - 17136*a^80 - 6048*a^76 + 3024*a^72 - 4536*a^68 + 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22449)/a^144)*t^5 + ((2016*a^204 - 1512*a^200 + 168*a^192 + 18144*a^188 - 65016*a^184 + 65016*a^180 + 51912*a^176 - 555408*a^172 + 715680*a^168 - 2029104*a^164 + 8481816*a^160 - 1542576*a^156 - 38264184*a^152 + 62637624*a^148 - 26911584*a^144 - 15769656*a^140 + 24697008*a^136 - 32813928*a^132 + 48401640*a^128 - 47118960*a^124 + 22057140*a^120 + 15539832*a^116 - 35120232*a^112 + 14938056*a^108 + 17504424*a^104 - 19161576*a^100 - 7819308*a^96 + 24211152*a^92 - 15733116*a^88 + 7275912*a^84 - 9064944*a^80 + 4148928*a^76 + 7590996*a^72 - 10214064*a^68 + 4367034*a^64 + 1021104*a^60 - 5400864*a^56 + 6758640*a^52 - 1948170*a^48 - 3391920*a^44 + 3877776*a^40 - 2057832*a^36 + 1030806*a^32 - 1199520*a^28 + 2076480*a^24 - 1890000*a^20 - 58590*a^16 + 1597680*a^12 - 1355004*a^8 + 487368*a^4 - 67284)/a^144)*t^4 + ((216*a^228 - 72*a^224 + 1512*a^208 - 6048*a^204 + 1872*a^200 + 4032*a^196 + 3654*a^192 - 54432*a^188 + 104328*a^184 - 107688*a^180 - 147420*a^176 + 734832*a^172 - 503964*a^168 + 2727648*a^164 - 8302644*a^160 - 1993824*a^156 + 29496096*a^152 - 34321896*a^148 + 10571736*a^144 + 5004720*a^140 - 8072064*a^136 + 14283864*a^132 - 22345092*a^128 + 25863264*a^124 - 16403688*a^120 - 3900960*a^116 + 12075084*a^112 + 2124192*a^108 - 17926272*a^104 + 12195792*a^100 + 9939321*a^96 - 20431152*a^92 + 11304720*a^88 - 4447296*a^84 + 4994766*a^80 + 1586592*a^76 - 9263072*a^72 + 5885712*a^68 - 359478*a^64 - 2390688*a^60 + 8754300*a^56 - 12358080*a^52 + 6247122*a^48 + 1014048*a^44 - 3329676*a^40 + 3264912*a^36 - 2712069*a^32 + 3976560*a^28 - 6822564*a^24 + 5518800*a^20 + 324576*a^16 - 3904992*a^12 + 2928240*a^8 - 945504*a^4 + 118124)/a^144)*t^3 + ((18*a^256 + 72*a^232 - 432*a^228 + 63*a^224 + 168*a^216 - 2772*a^208 + 6048*a^204 - 360*a^200 - 6912*a^196 - 7980*a^192 + 54432*a^188 - 77112*a^184 + 89376*a^180 + 139068*a^176 - 425808*a^172 + 161172*a^168 - 1804320*a^164 + 4068918*a^160 + 1949808*a^156 - 11683224*a^152 + 10662624*a^148 - 2992668*a^144 - 458640*a^140 + 950544*a^136 - 1688400*a^132 + 3778236*a^128 - 6995520*a^124 + 6626004*a^120 - 1819440*a^116 - 458136*a^112 - 2590896*a^108 + 5779368*a^104 - 3877272*a^100 - 2059155*a^96 + 5085360*a^92 - 2657340*a^88 - 127680*a^84 + 2014488*a^80 - 4914000*a^76 + 3886260*a^72 + 1572480*a^68 - 1814850*a^64 - 930240*a^60 - 3715560*a^56 + 10417680*a^52 - 10436244*a^48 + 6189120*a^44 - 851760*a^40 - 2876328*a^36 + 3322494*a^32 - 5511744*a^28 + 9497880*a^24 - 7166880*a^20 - 533736*a^16 + 4540032*a^12 - 3137616*a^8 + 940896*a^4 - 109584)/a^144)*t^2 + ((a^288 - 18*a^256 - 72*a^232 + 216*a^228 + 9*a^224 - 168*a^216 + 1260*a^208 - 2016*a^204 + 2880*a^196 + 4158*a^192 - 18144*a^188 + 22680*a^184 - 28560*a^180 - 45828*a^176 + 92160*a^172 - 20088*a^168 + 452592*a^164 - 794430*a^160 - 474768*a^156 + 1892520*a^152 - 1435392*a^148 + 373296*a^144 - 75600*a^140 + 13104*a^136 - 173376*a^132 + 568134*a^128 + 102816*a^124 - 1342656*a^120 + 1214640*a^116 - 275184*a^112 - 252672*a^108 + 491400*a^104 - 38808*a^100 - 1328922*a^96 + 1918224*a^92 - 1010520*a^88 + 620928*a^84 - 1473696*a^80 + 1658160*a^76 + 108920*a^72 - 1895040*a^68 + 525420*a^64 + 1975680*a^60 - 695520*a^56 - 3296160*a^52 + 5720400*a^48 - 4959360*a^44 + 1648080*a^40 + 1043280*a^36 - 1459080*a^32 + 2600640*a^28 - 4505760*a^24 + 3265920*a^20 + 272160*a^16 - 1935360*a^12 + 1270080*a^8 - 362880*a^4 + 40320)/a^144)*t]" ] }, "execution_count": 266, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# EGF of labeled digraphs,\n", "# with the marking variable t for strongly connected components (we take a parameter a = sqrt{sqrt{2}})\n", "dt = 1 / (exp_had_prod(exp(-t*scd),hatset,N))\n", "DT = [dt[i] * i.factorial() for i in srange(disp)]\n", "DT" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", " Asymptotics of satisfiable 2-CNFs\n", "" ] }, { "cell_type": "code", "execution_count": 267, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 1/2,\n", " (a^12 - 3/2*a^8 + a^4 - 1/4)/a^8,\n", " (a^36 - 3/2*a^28 - 2*a^24 + 3*a^20 + 9/4*a^16 - 3*a^12 - 3/4*a^8 + 3/2*a^4 - 3/8)/a^24,\n", " (a^72 - 2*a^60 - 3*a^52 + 7/2*a^48 - 3/2*a^44 + 12*a^40 - 8*a^36 - 12*a^28 + 19/2*a^24 + 21/4*a^16 - 3*a^12 - 21/4*a^8 + 9/2*a^4 - 15/16)/a^48,\n", " (a^120 - 5/2*a^104 - 5*a^92 + 10*a^88 - 15/2*a^84 - 3*a^80 + 30*a^76 - 30*a^72 + 25/2*a^68 + 35*a^64 - 375/4*a^60 + 65*a^56 - 30*a^52 - 20*a^48 + 105*a^44 - 345/4*a^40 + 65/2*a^36 + 5/16*a^32 - 40*a^28 + 165/4*a^24 - 45*a^20 + 375/8*a^16 - 255/8*a^8 + 75/4*a^4 - 105/32)/a^80,\n", " (a^180 - 3*a^160 - 15/2*a^144 + 15*a^140 - 15/4*a^136 - 10*a^132 + 105/2*a^124 - 127/2*a^120 - 15*a^116 + 90*a^112 + 75/2*a^108 - 240*a^104 + 171*a^100 + 60*a^96 - 405/4*a^92 - 615/2*a^88 + 2425/4*a^84 - 294*a^80 - 2145/16*a^76 + 225*a^72 + 300*a^68 - 1275/2*a^64 + 90*a^60 + 615/2*a^56 - 165/2*a^52 - 2675/8*a^48 + 1005/2*a^44 - 3315/16*a^40 - 165*a^36 + 1125/4*a^32 - 315/2*a^28 + 1245/8*a^24 - 1485/4*a^20 + 4365/16*a^16 + 405/4*a^12 - 6975/32*a^8 + 1575/16*a^4 - 945/64)/a^120,\n", " (a^252 - 7/2*a^228 - 21/2*a^208 + 21*a^204 - 21/4*a^200 - 35/2*a^192 + 105*a^184 - 245/2*a^180 - 105/4*a^176 + 42*a^172 + 983/8*a^168 + 105*a^164 - 630*a^160 + 490*a^156 + 1743/4*a^148 - 728*a^144 - 1155*a^140 + 5565/2*a^136 - 24605/16*a^132 + 441/2*a^128 - 1218*a^124 + 2233/4*a^120 + 18585/4*a^116 - 7455*a^112 + 29715/8*a^108 - 525/4*a^104 + 10353/8*a^100 - 11585/8*a^96 - 116445/32*a^92 + 25179/4*a^88 - 3787*a^84 + 15687/8*a^80 - 3465/2*a^76 + 14525/8*a^72 - 3255/4*a^68 - 16065/16*a^64 + 5845/4*a^60 - 38325/32*a^56 + 28245/16*a^52 - 237195/64*a^48 + 5670*a^44 - 33705/8*a^40 - 525*a^36 + 11655/4*a^32 - 7875/4*a^28 + 86415/32*a^24 - 31185/8*a^20 + 48825/32*a^16 + 23625/16*a^12 - 108045/64*a^8 + 19845/32*a^4 - 10395/128)/a^168,\n", " (a^336 - 4*a^308 - 14*a^284 + 28*a^280 - 7*a^276 - 28*a^264 + 168*a^256 - 168*a^252 - 77*a^248 + 84*a^244 - 21*a^240 + 252*a^236 + 210*a^232 - 1274*a^228 + 1531/2*a^224 + 280*a^220 + 735/2*a^216 + 350*a^212 - 3759/2*a^208 - 2275*a^204 + 53683/8*a^200 - 2932*a^196 - 420*a^192 + 238*a^188 - 7112*a^184 + 7266*a^180 + 12894*a^176 - 22239*a^172 + 7558*a^168 + 686*a^164 - 3745*a^160 + 62531/4*a^156 + 4872*a^152 - 56490*a^148 + 55636*a^144 - 8981*a^140 - 5313*a^136 + 18795/2*a^132 - 36848*a^128 + 66563/4*a^124 + 110509/2*a^120 - 565145/8*a^116 + 97125/4*a^112 + 145929/16*a^108 - 60529/2*a^104 + 55566*a^100 - 147119/4*a^96 - 30394*a^92 + 54922*a^88 - 28819*a^84 + 115465/4*a^80 - 33495*a^76 - 55895/4*a^72 + 122745/2*a^68 - 276675/8*a^64 - 82215/4*a^60 + 295785/16*a^56 + 42315/2*a^52 - 721035/16*a^48 + 51345*a^44 - 31815*a^40 - 11445/2*a^36 + 288435/16*a^32 - 79485/4*a^28 + 40320*a^24 - 162225/4*a^20 + 185535/32*a^16 + 147735/8*a^12 - 469665/32*a^8 + 72765/16*a^4 - 135135/256)/a^224,\n", " (a^432 - 9/2*a^400 - 18*a^372 + 36*a^368 - 9*a^364 - 42*a^348 + 252*a^340 - 252*a^336 - 63*a^332 + 63*a^328 - 63/2*a^324 + 504*a^316 + 315*a^312 - 2268*a^308 + 1386*a^304 - 42*a^300 + 2583/2*a^296 - 396*a^292 + 1187/2*a^288 - 3213*a^284 - 43533/8*a^280 + 15120*a^276 - 7560*a^272 + 2772*a^268 - 2310*a^264 - 5094*a^260 - 19827/2*a^256 + 11466*a^252 + 195993/4*a^248 - 73143*a^244 + 105903/4*a^240 - 15057/2*a^236 + 3753/16*a^232 - 45933/2*a^228 + 498195/4*a^224 - 50526*a^220 - 234528*a^216 + 289926*a^212 - 114030*a^208 + 51597*a^204 - 61425*a^200 + 553221/4*a^196 - 219681*a^192 - 1293075/8*a^188 + 2822337/4*a^184 - 9763005/16*a^180 + 231642*a^176 - 668421/4*a^172 + 402417/2*a^168 - 572733/2*a^164 + 1401939/4*a^160 + 108801*a^156 - 735777*a^152 + 5610591/8*a^148 - 259077*a^144 + 424305/8*a^140 - 742959/4*a^136 + 6857739/16*a^132 - 2476593/8*a^128 - 8248275/32*a^124 + 1015245/2*a^120 - 226485/2*a^116 - 5343219/16*a^112 + 523005*a^108 - 405531*a^104 - 130095/2*a^100 + 8012781/16*a^96 - 2389653/4*a^92 + 2166507/4*a^88 - 1937565/4*a^84 + 12406905/32*a^80 - 1890945/8*a^76 - 1486065/32*a^72 + 4843125/16*a^68 - 14763735/256*a^64 - 369495*a^60 + 1000755/8*a^56 + 1161405/2*a^52 - 31245795/32*a^48 + 812700*a^44 - 2070495/8*a^40 - 579285/4*a^36 + 5899635/32*a^32 - 348705*a^28 + 19496295/32*a^24 - 1686825/4*a^20 - 2520315/64*a^16 + 1845585/8*a^12 - 9074835/64*a^8 + 1216215/32*a^4 - 2027025/512)/a^288]" ] }, "execution_count": 267, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# IGF of satisfiable 2-CNF (we take a parameter a = sqrt{sqrt{2}})\n", "sat = g*exp_had_prod(exp(-scd/2),ddotset,N)\n", "SAT = [sat[i] * i.factorial() for i in srange(disp)]\n", "SAT" ] }, { "cell_type": "code", "execution_count": 268, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 1/2,\n", " 15/32,\n", " 799/1024,\n", " 1016571/524288,\n", " 28694311447/4026531840,\n", " 2034602766692687/49478023249920,\n", " 1115068294703296663717/2837267765243412480,\n", " 4795802950152171162502013473/743772721051969121157120,\n", " 163220487110350216972297148097903343/877390002843513269834754293760]" ] }, "execution_count": 268, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# IGF of satisfiable 2-CNF\n", "SATsb = [sat[i].subs(a=sqrt(sqrt(2))) for i in srange(disp)]\n", "SATsb" ] }, { "cell_type": "code", "execution_count": 269, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 1,\n", " 15,\n", " 2397,\n", " 3049713,\n", " 28694311447,\n", " 2034602766692687,\n", " 1115068294703296663717,\n", " 4795802950152171162502013473,\n", " 163220487110350216972297148097903343,\n", " 44164928411665942510460654486808636655906527,\n", " 95265878966579205334990504387281268292082212633510733,\n", " 1640680461860732121820130368901670064509937267196250426559284369,\n", " 225797401059395491388226781486494859477336476383420624645574889279833291655,\n", " 248449096473444603309693803281786498777604826860716062491779495426821364225621671245231,\n", " 2186249373678136497929092867348607033501533753383379007905060966599312858378097566945226476160367797,\n", " 153876507870619064316245403952108076113373975727815029772783646372965670866193991271419148214709969273888724439361,\n", " 86634689646631695792362052797375058873054204228562227682034608063501723861570442427435886561302900664407051568560067003936816351,\n", " 390191774172218805142782958749241933165131883043872896011181628217746441735276665013850091802123482655700143733528694386779753133685046362583743,\n", " 14058596106872313778290075591271893305303313138868553609649111587958343503927693511047338576434109913612519458536994817454174322323926252697999933306146761085405]" ] }, "execution_count": 269, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Numbers of satisfiable 2-CNF (see Section A.11)\n", "SATsub = [sat[i].subs(a=sqrt(sqrt(2))) * 2**(i*i) * i.factorial() for i in srange(N)]\n", "SATsub" ] }, { "cell_type": "code", "execution_count": 270, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " -1/2*a^4*w,\n", " (-1/2*a^8 + a^4 - 1/4)*w^2,\n", " ((-1/2*a^24 + 3*a^16 - 3*a^12 - 3/4*a^8 + 3/2*a^4 - 3/8)/a^12)*w^3,\n", " ((-1/2*a^48 + 4*a^36 + 3*a^32 - 18*a^28 + 11*a^24 + 21/4*a^16 - 3*a^12 - 21/4*a^8 + 9/2*a^4 - 15/16)/a^32)*w^4,\n", " ((-1/2*a^80 + 5*a^64 + 10*a^56 - 30*a^52 - 185/4*a^48 + 120*a^44 - 60*a^40 + 10*a^36 + 5*a^32 - 40*a^28 + 165/4*a^24 - 45*a^20 + 375/8*a^16 - 255/8*a^8 + 75/4*a^4 - 105/32)/a^60)*w^5,\n", " ((-1/2*a^120 + 6*a^100 + 15*a^88 - 35*a^84 - 3/2*a^80 - 180*a^76 + 195*a^72 + 540*a^68 - 885*a^64 + 360*a^60 + 105/4*a^56 - 165/2*a^52 - 1145/8*a^48 + 390*a^44 - 375/2*a^40 - 165*a^36 + 1125/4*a^32 - 315/2*a^28 + 1245/8*a^24 - 1485/4*a^20 + 4365/16*a^16 + 405/4*a^12 - 6975/32*a^8 + 1575/16*a^4 - 945/64)/a^96)*w^6,\n", " ((-1/2*a^168 + 7*a^144 + 21*a^128 - 63*a^124 + 133/4*a^120 - 315*a^112 + 210*a^108 - 315*a^104 + 2541*a^100 - 840*a^96 - 6300*a^92 + 30429/4*a^88 - 2632*a^84 - 63/8*a^80 - 630*a^76 + 2905/4*a^72 + 1785*a^68 - 11655/4*a^64 + 1505/2*a^60 + 2625/8*a^56 + 4305/4*a^52 - 57645/16*a^48 + 5670*a^44 - 33705/8*a^40 - 525*a^36 + 11655/4*a^32 - 7875/4*a^28 + 86415/32*a^24 - 31185/8*a^20 + 48825/32*a^16 + 23625/16*a^12 - 108045/64*a^8 + 19845/32*a^4 - 10395/128)/a^140)*w^7,\n", " ((-1/2*a^224 + 8*a^196 + 28*a^176 - 84*a^172 - 2*a^168 + 56*a^164 + 35*a^160 - 504*a^156 + 672*a^152 - 840*a^148 - 602*a^144 + 4200*a^140 - 840*a^136 + 10080*a^132 - 32263*a^128 - 5278*a^124 + 151459/2*a^120 - 70560*a^116 + 18900*a^112 + 924*a^108 - 1442*a^104 + 10206*a^100 - 12299/4*a^96 - 26194*a^92 + 31612*a^88 - 13069*a^84 + 14525/2*a^80 - 2310*a^76 - 26180*a^72 + 49560*a^68 - 84315/4*a^64 - 25515*a^60 + 76545/4*a^56 + 42315/2*a^52 - 721035/16*a^48 + 51345*a^44 - 31815*a^40 - 11445/2*a^36 + 288435/16*a^32 - 79485/4*a^28 + 40320*a^24 - 162225/4*a^20 + 185535/32*a^16 + 147735/8*a^12 - 469665/32*a^8 + 72765/16*a^4 - 135135/256)/a^192)*w^8,\n", " ((-1/2*a^288 + 9*a^256 + 36*a^232 - 108*a^228 - 9/4*a^224 + 84*a^216 - 630*a^208 + 1008*a^204 - 1476*a^196 - 2079*a^192 + 9072*a^188 - 11340*a^184 + 14280*a^180 + 22797*a^176 - 45720*a^172 + 20133/2*a^168 - 226548*a^164 + 794115/2*a^160 + 239652*a^156 - 949410*a^152 + 721728*a^148 - 184107*a^144 + 18900*a^140 - 3276*a^136 + 43344*a^132 - 142191*a^128 - 25200*a^124 + 1341585/4*a^120 - 303660*a^116 + 71631*a^112 + 60900*a^108 - 119196*a^104 - 13545*a^100 + 2708433/8*a^96 - 418572*a^92 + 714987/4*a^88 - 119385*a^84 + 5368545/16*a^80 - 402570*a^76 + 342615/4*a^72 + 261765*a^68 - 52920*a^64 - 369495*a^60 + 1000755/8*a^56 + 1161405/2*a^52 - 31245795/32*a^48 + 812700*a^44 - 2070495/8*a^40 - 579285/4*a^36 + 5899635/32*a^32 - 348705*a^28 + 19496295/32*a^24 - 1686825/4*a^20 - 2520315/64*a^16 + 1845585/8*a^12 - 9074835/64*a^8 + 1216215/32*a^4 - 2027025/512)/a^252)*w^9]" ] }, "execution_count": 270, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# CGF of satisfiable 2-CNF (we take a parameter a = sqrt{sqrt{2}})\n", "qsat = sat.subs(z = a^4*z*w)/g.subs(z = a^4*z*w)\n", "#qsat = sat.subs(z = a^4*z*w)(1-it.subs(z = a^4*z*w))\n", "QSAT = [qsat[i] * i.factorial() for i in srange(disp)]\n", "QSAT" ] }, { "cell_type": "code", "execution_count": 271, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " -1,\n", " -1/8,\n", " -67/384,\n", " -12559/98304,\n", " -8976361/125829120,\n", " -23458307761/773094113280,\n", " -225313054216027/22166154415964160,\n", " -8128654281172475359/2905362191609254379520,\n", " -1126432028071278245168401/1713652349303736855146004480,\n", " -609490791702860823991033024801/4492236814558787941553941984051200,\n", " -1300948293132139639890876373220523187/51814968810690751870631528968553182003200,\n", " -1574344512079993445231173637114114162540377/372561824472952222858281093392431689409285324800,\n", " -371776527820540023551336312444216873354327564436601/568800068430849309652190255623050928880268799281777868800,\n", " -50045450874559704310207287622920361216709858769477079698001/534401370097231836163198124334255840718194835882217801457126604800,\n", " -26910624372866634527569826069815524551813513831983494099510131606347/2151784132036127884172780683860159945357779974003324467381918669138296832000,\n", " -57839531795022957409894424509842783185090762125280894256621714102026438225599/36967369900587660611983076202239608010563248030032035130727837702723317578321625088000,\n", " -497065289633126177234465163032133187623765545934236343856613666428841801186631211664161/2699151960616699149663449185899614972140889138084908284219585509369871125303726071483063074816000,\n", " -2440457653421443452233348802023787785374804057512493143478061274477152181382794328186553045640823/119239913805768029203976814788537183770324148082202036990792381495739294237974850166469516706467761094656000,\n", " -213472924115436245566561438345276750724005535063026237950470817720641459275306199791426162487628753783595337/99074172382680921239169881081671528547597669014068086568472328304684691881500637998283251046810502577636646529794048000]" ] }, "execution_count": 271, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# CGF of satisfiable 2-CNF\n", "QSATsb = [qsat[i][i].subs(a=sqrt(sqrt(2))) for i in srange(N)]\n", "QSATsb" ] }, { "cell_type": "code", "execution_count": 272, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " -1/2*a^4*w,\n", " (-1/4*a^12 + 1/2*a^8 - 1/8*a^4)*w^2,\n", " (-1/12*a^24 + 1/2*a^16 - 1/2*a^12 - 1/8*a^8 + 1/4*a^4 - 1/16)*w^3,\n", " ((-1/48*a^48 + 1/6*a^36 + 1/8*a^32 - 3/4*a^28 + 11/24*a^24 + 7/32*a^16 - 1/8*a^12 - 7/32*a^8 + 3/16*a^4 - 5/128)/a^8)*w^4,\n", " ((-1/240*a^80 + 1/24*a^64 + 1/12*a^56 - 1/4*a^52 - 37/96*a^48 + a^44 - 1/2*a^40 + 1/12*a^36 + 1/24*a^32 - 1/3*a^28 + 11/32*a^24 - 3/8*a^20 + 25/64*a^16 - 17/64*a^8 + 5/32*a^4 - 7/256)/a^20)*w^5,\n", " ((-1/1440*a^120 + 1/120*a^100 + 1/48*a^88 - 7/144*a^84 - 1/480*a^80 - 1/4*a^76 + 13/48*a^72 + 3/4*a^68 - 59/48*a^64 + 1/2*a^60 + 7/192*a^56 - 11/96*a^52 - 229/1152*a^48 + 13/24*a^44 - 25/96*a^40 - 11/48*a^36 + 25/64*a^32 - 7/32*a^28 + 83/384*a^24 - 33/64*a^20 + 97/256*a^16 + 9/64*a^12 - 155/512*a^8 + 35/256*a^4 - 21/1024)/a^36)*w^6,\n", " ((-1/10080*a^168 + 1/720*a^144 + 1/240*a^128 - 1/80*a^124 + 19/2880*a^120 - 1/16*a^112 + 1/24*a^108 - 1/16*a^104 + 121/240*a^100 - 1/6*a^96 - 5/4*a^92 + 483/320*a^88 - 47/90*a^84 - 1/640*a^80 - 1/8*a^76 + 83/576*a^72 + 17/48*a^68 - 37/64*a^64 + 43/288*a^60 + 25/384*a^56 + 41/192*a^52 - 183/256*a^48 + 9/8*a^44 - 107/128*a^40 - 5/48*a^36 + 37/64*a^32 - 25/64*a^28 + 823/1536*a^24 - 99/128*a^20 + 155/512*a^16 + 75/256*a^12 - 343/1024*a^8 + 63/512*a^4 - 33/2048)/a^56)*w^7,\n", " ((-1/80640*a^224 + 1/5040*a^196 + 1/1440*a^176 - 1/480*a^172 - 1/20160*a^168 + 1/720*a^164 + 1/1152*a^160 - 1/80*a^156 + 1/60*a^152 - 1/48*a^148 - 43/2880*a^144 + 5/48*a^140 - 1/48*a^136 + 1/4*a^132 - 4609/5760*a^128 - 377/2880*a^124 + 21637/11520*a^120 - 7/4*a^116 + 15/32*a^112 + 11/480*a^108 - 103/2880*a^104 + 81/320*a^100 - 1757/23040*a^96 - 1871/2880*a^92 + 1129/1440*a^88 - 1867/5760*a^84 + 415/2304*a^80 - 11/192*a^76 - 187/288*a^72 + 59/48*a^68 - 803/1536*a^64 - 81/128*a^60 + 243/512*a^56 + 403/768*a^52 - 2289/2048*a^48 + 163/128*a^44 - 101/128*a^40 - 109/768*a^36 + 2747/6144*a^32 - 757/1536*a^28 + a^24 - 515/512*a^20 + 589/4096*a^16 + 469/1024*a^12 - 1491/4096*a^8 + 231/2048*a^4 - 429/32768)/a^80)*w^8,\n", " ((-1/725760*a^288 + 1/40320*a^256 + 1/10080*a^232 - 1/3360*a^228 - 1/161280*a^224 + 1/4320*a^216 - 1/576*a^208 + 1/360*a^204 - 41/10080*a^196 - 11/1920*a^192 + 1/40*a^188 - 1/32*a^184 + 17/432*a^180 + 2533/40320*a^176 - 127/1008*a^172 + 2237/80640*a^168 - 899/1440*a^164 + 2521/2304*a^160 + 317/480*a^156 - 1507/576*a^152 + 179/90*a^148 - 8767/17280*a^144 + 5/96*a^140 - 13/1440*a^136 + 43/360*a^132 - 2257/5760*a^128 - 5/72*a^124 + 4259/4608*a^120 - 241/288*a^116 + 379/1920*a^112 + 145/864*a^108 - 473/1440*a^104 - 43/1152*a^100 + 42991/46080*a^96 - 1661/1440*a^92 + 1261/2560*a^88 - 379/1152*a^84 + 5681/6144*a^80 - 71/64*a^76 + 3263/13824*a^72 + 277/384*a^68 - 7/48*a^64 - 391/384*a^60 + 353/1024*a^56 + 1229/768*a^52 - 99193/36864*a^48 + 215/96*a^44 - 2191/3072*a^40 - 613/1536*a^36 + 2081/4096*a^32 - 123/128*a^28 + 6877/4096*a^24 - 595/512*a^20 - 889/8192*a^16 + 651/1024*a^12 - 3201/8192*a^8 + 429/4096*a^4 - 715/65536)/a^108)*w^9]" ] }, "execution_count": 272, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of satisfiable 2-CNF (we take a parameter a = sqrt{sqrt{2}})\n", "QSATzero = [qsat[i] * a^(2*i*(i-1)) for i in srange(disp)]\n", "QSATzero" ] }, { "cell_type": "code", "execution_count": 273, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -1, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -1/4, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, -67/48, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, -12559/1536, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, -8976361/122880, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, -23458307761/23592960, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, -225313054216027/10569646080, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, -8128654281172475359/10823317585920, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, -1126432028071278245168401/24936923717959680]]" ] }, "execution_count": 273, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of satisfiable 2-CNF (see Section A.11)\n", "QSATzeroMatrix = [[QSATzero[i][j].subs(a=sqrt(sqrt(2))) for j in srange(disp)] for i in srange(disp)]\n", "QSATzeroMatrix" ] }, { "cell_type": "code", "execution_count": 274, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[-1,\n", " 1,\n", " 1/4,\n", " 67/48,\n", " 12559/1536,\n", " 8976361/122880,\n", " 23458307761/23592960,\n", " 225313054216027/10569646080,\n", " 8128654281172475359/10823317585920,\n", " 1126432028071278245168401/24936923717959680]" ] }, "execution_count": 274, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Verification\n", "QSATzeroVer = [2**(i*(i+1)/2) * (sum(binomial(i,j)*SATsub[j]*ITsub[i-j]/2**(j*j) for j in srange(i)) - SATsub[i]/2**(i*i)) / i.factorial() for i in srange(disp)]\n", "QSATzeroVer" ] }, { "cell_type": "code", "execution_count": 275, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " -1,\n", " -1,\n", " -67,\n", " -12559,\n", " -8976361,\n", " -23458307761,\n", " -225313054216027,\n", " -8128654281172475359,\n", " -1126432028071278245168401]" ] }, "execution_count": 275, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients bar(s_m) (see Section A.11)\n", "QSATzeroLine = [QSATzeroMatrix[i][i] * 2**(i*(i-1)/2) * i.factorial() for i in srange(disp)]\n", "QSATzeroLine" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", " Asymptotics of contradictory strongly connected implication digraphs\n", "" ] }, { "cell_type": "code", "execution_count": 276, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " 0,\n", " a^16 - 4*a^12 + 6*a^8 - 4*a^4 + 1,\n", " a^48 - 8*a^36 - 6*a^32 + 36*a^28 - 20*a^24 - 21*a^16 + 12*a^12 + 24*a^8 - 24*a^4 + 6,\n", " a^96 - 24*a^72 + 72*a^60 + 144*a^56 - 384*a^52 + 204*a^48 - 96*a^36 - 75*a^32 + 456*a^28 - 328*a^24 + 96*a^20 - 204*a^16 + 48*a^12 + 240*a^8 - 192*a^4 + 42,\n", " a^160 - 10*a^128 - 32*a^120 - 40*a^108 + 320*a^104 - 80*a^100 + 645*a^96 - 800*a^92 - 2880*a^88 + 4800*a^84 - 1904*a^80 - 120*a^72 - 170*a^64 + 400*a^60 + 360*a^56 - 880*a^52 + 2240*a^48 - 4120*a^44 + 3840*a^40 - 3200*a^36 + 490*a^32 + 3520*a^28 - 3760*a^24 + 3840*a^20 - 3780*a^16 - 240*a^12 + 3120*a^8 - 1920*a^4 + 360,\n", " a^240 - 12*a^200 - 64*a^180 - 60*a^172 + 120*a^168 + 582*a^160 - 160*a^156 - 240*a^152 + 2880*a^148 - 3520*a^144 + 1920*a^140 - 17760*a^136 + 4800*a^132 + 57540*a^128 - 69120*a^124 + 22880*a^120 - 15*a^112 - 180*a^108 + 1860*a^104 - 864*a^100 + 3860*a^96 - 4800*a^92 - 17880*a^88 + 29760*a^84 - 9768*a^80 + 9720*a^76 - 11200*a^72 - 25440*a^68 + 26400*a^64 + 14640*a^60 - 26880*a^56 + 17280*a^52 + 15590*a^48 - 53640*a^44 + 48840*a^40 - 15360*a^36 - 18750*a^32 + 27600*a^28 - 37200*a^24 + 72000*a^20 - 52920*a^16 - 20640*a^12 + 47520*a^8 - 23040*a^4 + 3720,\n", " a^336 - 14*a^288 - 212*a^252 + 168*a^248 + 7*a^240 + 1064*a^228 + 1680*a^220 - 2912*a^216 + 5376*a^212 - 10752*a^208 + 13440*a^204 + 9996*a^200 - 87360*a^196 + 49280*a^192 - 120960*a^188 + 423360*a^184 + 120512*a^180 - 1209621*a^176 + 1128624*a^172 - 321740*a^168 + 4074*a^160 - 1120*a^156 - 1680*a^152 + 20160*a^148 - 25361*a^144 + 12600*a^140 - 122808*a^136 + 31192*a^132 + 402990*a^128 - 476196*a^124 + 147504*a^120 + 31920*a^116 - 6930*a^112 + 40320*a^108 - 17640*a^104 - 563136*a^100 + 813190*a^96 + 263760*a^92 - 1102416*a^88 + 387072*a^84 + 589218*a^80 - 430080*a^76 + 3360*a^72 - 337680*a^68 + 345240*a^64 + 295400*a^60 - 493920*a^56 + 164640*a^52 + 535430*a^48 - 1400280*a^44 + 1368360*a^40 - 141120*a^36 - 729960*a^32 + 655200*a^28 - 1075200*a^24 + 1612800*a^20 - 682920*a^16 - 655200*a^12 + 816480*a^8 - 322560*a^4 + 45360,\n", " a^448 - 16*a^392 - 112*a^348 + 224*a^344 - 248*a^336 - 448*a^316 + 5760*a^308 - 2688*a^304 - 1120*a^296 - 1792*a^292 + 12432*a^288 - 19712*a^284 + 6720*a^280 - 11648*a^276 + 84224*a^272 - 215040*a^268 + 250880*a^264 - 412160*a^260 - 609308*a^256 + 1933776*a^252 - 858928*a^248 + 5214720*a^244 - 9461704*a^240 - 7526400*a^236 + 27095040*a^232 - 20635328*a^228 + 5161088*a^224 + 13440*a^220 - 22960*a^216 + 41664*a^212 - 84728*a^208 + 107520*a^204 + 79968*a^200 - 698688*a^196 + 387373*a^192 - 960848*a^188 + 3378144*a^184 + 984256*a^180 - 9713312*a^176 + 9120160*a^172 - 2646320*a^168 - 108416*a^164 + 468272*a^160 - 647472*a^156 + 868896*a^152 - 1866368*a^148 + 736176*a^144 + 7042560*a^140 - 15279488*a^136 + 7384832*a^132 + 9725604*a^128 - 6070624*a^124 - 14466144*a^120 + 23318400*a^116 - 18246760*a^112 + 13834912*a^108 - 9977856*a^104 - 3750656*a^100 + 11145792*a^96 + 5382048*a^92 - 17764992*a^88 + 6087424*a^84 + 7614768*a^80 - 12116160*a^76 + 23461760*a^72 - 35548800*a^68 + 18039210*a^64 + 13125280*a^60 - 14554400*a^56 - 4968320*a^52 + 19766880*a^48 - 29796480*a^44 + 22915200*a^40 + 1774080*a^36 - 12779760*a^32 + 16356480*a^28 - 34742400*a^24 + 36933120*a^20 - 6189120*a^16 - 18144000*a^12 + 15563520*a^8 - 5160960*a^4 + 640080,\n", " a^576 - 18*a^512 - 144*a^460 + 288*a^456 + 9*a^448 - 512*a^432 - 672*a^420 + 4032*a^412 - 4032*a^408 + 6912*a^400 - 2160*a^392 + 11520*a^380 + 44928*a^376 - 151680*a^372 + 48384*a^368 + 126336*a^360 + 64512*a^356 - 370980*a^352 + 402336*a^348 - 361008*a^344 + 32256*a^340 - 3410616*a^336 + 7289856*a^332 - 10450944*a^328 + 9945600*a^324 + 28062720*a^320 - 39678912*a^316 + 33546240*a^312 - 193484160*a^308 + 199318392*a^304 + 294172704*a^300 - 650289024*a^296 + 418021632*a^292 - 92785220*a^288 - 177408*a^284 + 60480*a^280 - 104160*a^276 + 758520*a^272 - 1930752*a^268 + 2223360*a^264 - 3654720*a^260 - 5512950*a^256 + 17404992*a^252 - 7730352*a^248 + 46894176*a^244 - 85002120*a^240 - 67959360*a^236 + 244180656*a^232 - 186475776*a^228 + 48008556*a^224 - 2689344*a^220 + 2806944*a^216 + 3040128*a^212 - 16153200*a^208 + 24422160*a^204 - 38844288*a^200 + 68186592*a^196 + 5032734*a^192 - 247433760*a^188 + 390009312*a^184 - 209147904*a^180 - 65372184*a^176 + 115607232*a^172 - 47115648*a^168 + 208276992*a^164 - 278544672*a^160 - 292634496*a^156 + 818127072*a^152 - 541437120*a^148 + 108263624*a^144 - 62991936*a^140 + 69600384*a^136 - 78161664*a^132 + 196681212*a^128 - 86958144*a^124 - 247193184*a^120 + 389975040*a^116 - 354718224*a^112 + 327826688*a^108 - 216770400*a^104 + 85865472*a^100 - 280442232*a^96 + 639059904*a^92 - 633941280*a^88 + 392067648*a^84 - 310420656*a^80 + 213927840*a^76 + 268755200*a^72 - 745799040*a^68 + 379821960*a^64 + 330140160*a^60 - 118782720*a^56 - 754064640*a^52 + 1272731040*a^48 - 1113073920*a^44 + 384531840*a^40 + 229340160*a^36 - 329162400*a^32 + 641088000*a^28 - 1151781120*a^24 + 844784640*a^20 + 65499840*a^16 - 496177920*a^12 + 325866240*a^8 - 92897280*a^4 + 10281600]" ] }, "execution_count": 276, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# EGF of contradictory strongly connected implication digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "cscc = log(exp_had_prod(d/g.subs(z=2*z),d,N)) + scd.subs(z=2*z)/2\n", "CSCC = [cscc[i] * i.factorial() for i in srange(disp)]\n", "CSCC" ] }, { "cell_type": "code", "execution_count": 277, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " 0,\n", " 1,\n", " 1606,\n", " 12864042,\n", " 1035697286504,\n", " 1137724245192445576,\n", " 19275699325699284398997808,\n", " 5187221831835207014808821762704464,\n", " 22294619997952835408033533526860039478469248,\n", " 1532378620903971051220118091442016459229955421936318080,\n", " 1684961314076455535365379591999039797953859605423390876101655503104,\n", " 29642605227454854630551408130507373873482741310983797918043445583780311814302464,\n", " 8343686428662293190981251306447000897822543196781971769510910299825116404717213746936206525440,\n", " 37576665646168253166866482504052164868974270211375861912370715699343527242379360781177046860802446211634115584,\n", " 2707684945557451236582206955141024326876877071965141524178301505534293508529886661571956358105100236930563137233767989679806464,\n", " 3121748457280635564865428301743914485519056577281653558208335774024045241858742454148748399032510665780515208957477600857184656604559187452143616,\n", " 57586096114287444134995809173660516831168083903214752826003829800109384494951683651682173505345824514486845196932414284209416836541778724905135505112784920768053248,\n", " 16996415734520969346976316634185422653151718716397360127143532564908212078333732597468381359902999464657535664852571684255507969168905044622778371903233605242287145369230273632552517632,\n", " 80263304117426476159047797543163947531906894435969580136809190481139563434236982122744001003209174414285402067740650764605666397087324831155987004502670832435475322879659307954415090718150785028137648193536]" ] }, "execution_count": 277, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Numbers of contradictory strongly connected implication digraphs (see Section A.12)\n", "CSCCsub = [cscc[i].subs(a=sqrt(sqrt(2))) * i.factorial() for i in srange(N)]\n", "CSCCsub" ] }, { "cell_type": "code", "execution_count": 278, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 0,\n", " -2*a^11*w,\n", " 0,\n", " (-12*a^30 + 24*a^26)*w^2 + 12*a^14*w,\n", " 0,\n", " (-120*a^57 + 720*a^49 - 720*a^45)*w^3 - 360*a^25*w^2,\n", " 0,\n", " (-1680*a^92 + 13440*a^80 + 10080*a^76 - 60480*a^72 + 40320*a^68)*w^4 + (-10080*a^44 + 20160*a^40)*w^3 + (-10080*a^36 + 60480*a^32 - 105840*a^28 + 80640*a^24 - 20160*a^20)*w^2,\n", " 0,\n", " (-30240*a^135 + 302400*a^119 + 604800*a^111 - 1814400*a^107 - 2721600*a^103 + 7257600*a^99 - 3628800*a^95)*w^5 + (-302400*a^71 + 1814400*a^63 - 1814400*a^59)*w^4 + (907200*a^47 - 5443200*a^43 + 9525600*a^39 - 7257600*a^35 + 1814400*a^31)*w^3,\n", " 0,\n", " (-665280*a^186 + 7983360*a^166 + 19958400*a^154 - 46569600*a^150 - 239500800*a^142 + 259459200*a^138 + 718502400*a^134 - 1197504000*a^130 + 479001600*a^126)*w^6 + (-9979200*a^106 + 79833600*a^94 + 59875200*a^90 - 359251200*a^86 + 239500800*a^82)*w^5 + (59875200*a^66 - 479001600*a^62 + 1347192000*a^58 - 1736380800*a^54 + 1077753600*a^50 - 239500800*a^46)*w^4 + (-79833600*a^78 + 678585600*a^66 + 479001600*a^62 - 3113510400*a^58 + 1836172800*a^54 + 59875200*a^50 + 1796256000*a^46 - 1526817600*a^42 - 1437004800*a^38 + 1796256000*a^34 - 479001600*a^30)*w^3,\n", " 0,\n", " (-17297280*a^245 + 242161920*a^221 + 726485760*a^205 - 2179457280*a^201 + 1210809600*a^197 - 10897286400*a^189 + 7264857600*a^185 - 10897286400*a^181 + 87178291200*a^177 - 29059430400*a^173 - 217945728000*a^169 + 261534873600*a^165 - 87178291200*a^161)*w^7 + (-363242880*a^149 + 3632428800*a^133 + 7264857600*a^125 - 21794572800*a^121 - 32691859200*a^117 + 87178291200*a^113 - 43589145600*a^109)*w^6 + (3632428800*a^93 - 21794572800*a^89 + 16345929600*a^85 + 123502579200*a^81 - 352345593600*a^77 + 403199596800*a^73 - 217945728000*a^69 + 43589145600*a^65)*w^5 + (14529715200*a^89 - 123502579200*a^77 - 87178291200*a^73 + 566658892800*a^69 - 334183449600*a^65 - 10897286400*a^61 - 326918592000*a^57 + 277880803200*a^53 + 261534873600*a^49 - 326918592000*a^45 + 87178291200*a^41)*w^4,\n", " 0,\n", " (-518918400*a^312 + 8302694400*a^284 + 29059430400*a^264 - 87178291200*a^260 + 58118860800*a^252 + 36324288000*a^248 - 523069747200*a^244 + 697426329600*a^240 - 871782912000*a^236 - 653837184000*a^232 + 4358914560000*a^228 - 871782912000*a^224 + 10461394944000*a^220 - 33563642112000*a^216 - 5230697472000*a^212 + 78460462080000*a^208 - 73229764608000*a^204 + 20922789888000*a^200)*w^8 + (-14529715200*a^200 + 174356582400*a^180 + 435891456000*a^168 - 1017080064000*a^164 - 5230697472000*a^156 + 5666588928000*a^152 + 15692092416000*a^148 - 26153487360000*a^144 + 10461394944000*a^140)*w^7 + (217945728000*a^128 - 1307674368000*a^124 + 2288430144000*a^120 - 3487131648000*a^116 + 9589612032000*a^112 - 2615348736000*a^108 - 52089028992000*a^104 + 120741933312000*a^100 - 120306041856000*a^96 + 57537672192000*a^92 - 10461394944000*a^88)*w^6 + (1743565824000*a^108 - 3487131648000*a^104 - 14820309504000*a^96 + 19179224064000*a^92 + 88921857024000*a^88 - 176100148224000*a^84 + 78896353536000*a^80 - 36614882304000*a^76 + 111806158464000*a^72 - 35307207936000*a^68 - 101998600704000*a^64 + 88921857024000*a^60 - 20922789888000*a^56)*w^5 + (-871782912000*a^136 + 20922789888000*a^112 + 435891456000*a^104 - 62768369664000*a^100 - 125536739328000*a^96 + 329533940736000*a^92 - 180459062784000*a^88 + 15692092416000*a^84 + 6102480384000*a^80 + 94152554496000*a^76 - 11115232128000*a^72 - 360046342656000*a^68 + 336290258304000*a^64 - 155613249792000*a^60 + 298640133792000*a^56 - 209227898880000*a^52 - 98075577600000*a^48 + 135998134272000*a^44 - 33999533568000*a^40)*w^4,\n", " 0,\n", " (-17643225600*a^387 + 317578060800*a^355 + 1270312243200*a^331 - 3810936729600*a^327 + 2964061900800*a^315 - 22230464256000*a^307 + 35568742809600*a^303 - 53353114214400*a^295 - 73360532044800*a^291 + 320118685286400*a^287 - 400148356608000*a^283 + 503890523136000*a^279 + 800296713216000*a^275 - 1600593426432000*a^271 + 355687428096000*a^267 - 8002967132160000*a^263 + 14005192481280000*a^259 + 8536498274304000*a^255 - 33612461955072000*a^251 + 25609494822912000*a^247 - 6402373705728000*a^243)*w^9 + (-635156121600*a^259 + 8892185702400*a^235 + 26676557107200*a^219 - 80029671321600*a^215 + 44460928512000*a^211 - 400148356608000*a^203 + 266765571072000*a^199 - 400148356608000*a^195 + 3201186852864000*a^191 - 1067062284288000*a^187 - 8002967132160000*a^183 + 9603560558592000*a^179 - 3201186852864000*a^175)*w^8 + (13338278553600*a^171 - 80029671321600*a^167 + 140051924812800*a^163 - 106706228428800*a^159 - 106706228428800*a^155 + 800296713216000*a^151 - 1667284819200000*a^147 + 3467952423936000*a^143 - 6669139276800000*a^139 + 133382785536000*a^135 + 26476482928896000*a^131 - 51218989645824000*a^127 + 44816615940096000*a^123 - 19207121117184000*a^119 + 3201186852864000*a^115)*w^7 + (177843714048000*a^135 - 1067062284288000*a^127 - 444609285120000*a^123 - 1067062284288000*a^119 + 16005934264320000*a^115 - 6758061133824000*a^111 - 48151185578496000*a^107 + 62156378059776000*a^103 - 20340874794240000*a^99 + 26409791536128000*a^95 - 48417951149568000*a^91 + 2267507354112000*a^87 + 43216022513664000*a^83 - 30411275102208000*a^79 + 6402373705728000*a^75)*w^6 + (266765571072000*a^147 - 6402373705728000*a^123 - 133382785536000*a^115 + 19207121117184000*a^111 + 38414242234368000*a^107 - 100837385865216000*a^103 + 55220473211904000*a^99 - 4801780279296000*a^95 - 1867358997504000*a^91 - 28810681675776000*a^87 + 3401261031168000*a^83 + 110174180852736000*a^79 - 102904819041024000*a^75 + 47617654436352000*a^71 - 91383880940352000*a^67 + 64023737057280000*a^63 + 30011126745600000*a^59 - 41615429087232000*a^55 + 10403857271808000*a^51)*w^5,\n", " 0]" ] }, "execution_count": 278, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# CGF of contradictory strongly connected implication digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "qcscc = exp(scd.subs(z = a^(14)*z^4*w)/2 - cscc.subs(z = a^(10)*z^4*w)) * exp_had_prod(PhiTwoTwoFour,1/g.subs(z = a^(10)*z^2*w),N)\n", "QCSCC = [qcscc[i] * i.factorial() for i in srange(N)]\n", "QCSCC" ] }, { "cell_type": "code", "execution_count": 279, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 0,\n", " -a^12*w,\n", " 0,\n", " (-1/2*a^36 + a^32)*w^2 + 1/2*a^20*w,\n", " 0,\n", " (-1/6*a^72 + a^64 - a^60)*w^3 - 1/2*a^40*w^2,\n", " 0,\n", " (-1/24*a^120 + 1/3*a^108 + 1/4*a^104 - 3/2*a^100 + a^96)*w^4 + (-1/4*a^72 + 1/2*a^68)*w^3 + (-1/4*a^64 + 3/2*a^60 - 21/8*a^56 + 2*a^52 - 1/2*a^48)*w^2,\n", " 0,\n", " (-1/120*a^180 + 1/12*a^164 + 1/6*a^156 - 1/2*a^152 - 3/4*a^148 + 2*a^144 - a^140)*w^5 + (-1/12*a^116 + 1/2*a^108 - 1/2*a^104)*w^4 + (1/4*a^92 - 3/2*a^88 + 21/8*a^84 - 2*a^80 + 1/2*a^76)*w^3,\n", " 0,\n", " (-1/720*a^252 + 1/60*a^232 + 1/24*a^220 - 7/72*a^216 - 1/2*a^208 + 13/24*a^204 + 3/2*a^200 - 5/2*a^196 + a^192)*w^6 + (-1/48*a^172 + 1/6*a^160 + 1/8*a^156 - 3/4*a^152 + 1/2*a^148)*w^5 + (1/8*a^132 - a^128 + 45/16*a^124 - 29/8*a^120 + 9/4*a^116 - 1/2*a^112)*w^4 + (-1/6*a^144 + 17/12*a^132 + a^128 - 13/2*a^124 + 23/6*a^120 + 1/8*a^116 + 15/4*a^112 - 51/16*a^108 - 3*a^104 + 15/4*a^100 - a^96)*w^3,\n", " 0,\n", " (-1/5040*a^336 + 1/360*a^312 + 1/120*a^296 - 1/40*a^292 + 1/72*a^288 - 1/8*a^280 + 1/12*a^276 - 1/8*a^272 + a^268 - 1/3*a^264 - 5/2*a^260 + 3*a^256 - a^252)*w^7 + (-1/240*a^240 + 1/24*a^224 + 1/12*a^216 - 1/4*a^212 - 3/8*a^208 + a^204 - 1/2*a^200)*w^6 + (1/24*a^184 - 1/4*a^180 + 3/16*a^176 + 17/12*a^172 - 97/24*a^168 + 37/8*a^164 - 5/2*a^160 + 1/2*a^156)*w^5 + (1/6*a^180 - 17/12*a^168 - a^164 + 13/2*a^160 - 23/6*a^156 - 1/8*a^152 - 15/4*a^148 + 51/16*a^144 + 3*a^140 - 15/4*a^136 + a^132)*w^4,\n", " 0,\n", " (-1/40320*a^432 + 1/2520*a^404 + 1/720*a^384 - 1/240*a^380 + 1/360*a^372 + 1/576*a^368 - 1/40*a^364 + 1/30*a^360 - 1/24*a^356 - 1/32*a^352 + 5/24*a^348 - 1/24*a^344 + 1/2*a^340 - 77/48*a^336 - 1/4*a^332 + 15/4*a^328 - 7/2*a^324 + a^320)*w^8 + (-1/1440*a^320 + 1/120*a^300 + 1/48*a^288 - 7/144*a^284 - 1/4*a^276 + 13/48*a^272 + 3/4*a^268 - 5/4*a^264 + 1/2*a^260)*w^7 + (1/96*a^248 - 1/16*a^244 + 7/64*a^240 - 1/6*a^236 + 11/24*a^232 - 1/8*a^228 - 239/96*a^224 + 277/48*a^220 - 23/4*a^216 + 11/4*a^212 - 1/2*a^208)*w^6 + (1/12*a^228 - 1/6*a^224 - 17/24*a^216 + 11/12*a^212 + 17/4*a^208 - 101/12*a^204 + 181/48*a^200 - 7/4*a^196 + 171/32*a^192 - 27/16*a^188 - 39/8*a^184 + 17/4*a^180 - a^176)*w^5 + (-1/24*a^256 + a^232 + 1/48*a^224 - 3*a^220 - 6*a^216 + 63/4*a^212 - 69/8*a^208 + 3/4*a^204 + 7/24*a^200 + 9/2*a^196 - 17/32*a^192 - 413/24*a^188 + 1543/96*a^184 - 119/16*a^180 + 1827/128*a^176 - 10*a^172 - 75/16*a^168 + 13/2*a^164 - 13/8*a^160)*w^4,\n", " 0,\n", " (-1/362880*a^540 + 1/20160*a^508 + 1/5040*a^484 - 1/1680*a^480 + 1/2160*a^468 - 1/288*a^460 + 1/180*a^456 - 1/120*a^448 - 11/960*a^444 + 1/20*a^440 - 1/16*a^436 + 17/216*a^432 + 1/8*a^428 - 1/4*a^424 + 1/18*a^420 - 5/4*a^416 + 35/16*a^412 + 4/3*a^408 - 21/4*a^404 + 4*a^400 - a^396)*w^9 + (-1/10080*a^412 + 1/720*a^388 + 1/240*a^372 - 1/80*a^368 + 1/144*a^364 - 1/16*a^356 + 1/24*a^352 - 1/16*a^348 + 1/2*a^344 - 1/6*a^340 - 5/4*a^336 + 3/2*a^332 - 1/2*a^328)*w^8 + (1/480*a^324 - 1/80*a^320 + 7/320*a^316 - 1/60*a^312 - 1/60*a^308 + 1/8*a^304 - 25/96*a^300 + 13/24*a^296 - 25/24*a^292 + 1/48*a^288 + 397/96*a^284 - 8*a^280 + 7*a^276 - 3*a^272 + 1/2*a^268)*w^7 + (1/36*a^288 - 1/6*a^280 - 5/72*a^276 - 1/6*a^272 + 5/2*a^268 - 19/18*a^264 - 361/48*a^260 + 233/24*a^256 - 305/96*a^252 + 33/8*a^248 - 121/16*a^244 + 17/48*a^240 + 27/4*a^236 - 19/4*a^232 + a^228)*w^6 + (1/24*a^300 - a^276 - 1/48*a^268 + 3*a^264 + 6*a^260 - 63/4*a^256 + 69/8*a^252 - 3/4*a^248 - 7/24*a^244 - 9/2*a^240 + 17/32*a^236 + 413/24*a^232 - 1543/96*a^228 + 119/16*a^224 - 1827/128*a^220 + 10*a^216 + 75/16*a^212 - 13/2*a^208 + 13/8*a^204)*w^5,\n", " 0]" ] }, "execution_count": 279, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of contradictory strongly connected implication digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "QCSCCzero = [qcscc[i] * a^(i*(i-1)/2) for i in srange(N)]\n", "QCSCCzero" ] }, { "cell_type": "code", "execution_count": 280, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -512, -32768/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 4096, 0, -16777216, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -524288,\n", " -33554432/3,\n", " -2336462209024/15,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -4278190080,\n", " 0,\n", " -68719476736,\n", " -8725724278030336,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 2190433320960,\n", " -549755813888/3,\n", " -38280596832649216/15,\n", " -967005994459965260038144/315,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " -1767096605254615040/3,\n", " 0,\n", " -4503599627370496,\n", " -571849066284996100096,\n", " -106375800719530450340745838592/15,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 3619013847561451601920/3,\n", " 12249790986447749120,\n", " -10035028776097996079104/15,\n", " -253494819411713133127439220736/315,\n", " -317041834428250708890138228122605060096/2835,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]" ] }, "execution_count": 280, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of contradictory strongly connected implication digraphs (Table 23)\n", "QCSCCzeroMatrix = [[QCSCCzero[i][j].subs(a=sqrt(sqrt(2))) for j in srange(N)] for i in srange(N)]\n", "QCSCCzeroMatrix" ] }, { "cell_type": "code", "execution_count": 169, "metadata": {}, "outputs": [], "source": [ "n = var('n')" ] }, { "cell_type": "code", "execution_count": 170, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-512/3*(64*n - 125)*(n - 1)*n" ] }, "execution_count": 170, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Polynomial w_3(n) (see Section A.12)\n", "expr3 = -512*n*(n-1)-32768*n*(n-1)*(n-2)/3\n", "expr3.factor()" ] }, { "cell_type": "code", "execution_count": 171, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-4096*(4096*n^2 - 20480*n + 24575)*(n - 1)*n" ] }, "execution_count": 171, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Polynomial w_4(n) (see Section A.12)\n", "expr4 = 4096*n*(n-1)-16777216*n*(n-1)*(n-2)*(n-3)\n", "expr4.factor()" ] }, { "cell_type": "code", "execution_count": 172, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-524288/15*(4456448*n^2 - 31194816*n + 53476431)*(n - 1)*(n - 2)*n" ] }, "execution_count": 172, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Polynomial w_5(n) (see Section A.12)\n", "expr5 = -524288*n*(n-1)*(n-2)-33554432*n*(n-1)*(n-2)*(n-3)/3-2336462209024*n*(n-1)*(n-2)*(n-3)*(n-4)/15\n", "expr5.factor()" ] }, { "cell_type": "code", "execution_count": 173, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-16777216*(520097792*n^2 - 6241153024*n + 14042579199)*(n - 1)*(n - 2)*n" ] }, "execution_count": 173, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Polynomial w_6(n) (see Section A.12)\n", "expr6 = -4278190080*n*(n-1)*(n-2)-68719476736*n*(n-1)*(n-2)*(n-3)*(n-4)-8725724278030336*n*(n-1)*(n-2)*(n-3)*(n-4)(n-5)\n", "expr6.factor()\n" ] }, { "cell_type": "code", "execution_count": 174, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1024" ] }, "execution_count": 174, "metadata": {}, "output_type": "execute_result" } ], "source": [ "524288/512" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", " Asymptotics of 2-CNF with different types of strongly connected components\n", "" ] }, { "cell_type": "code", "execution_count": 281, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 1/2*t,\n", " ((a^8 - a^4 + 1/4)/a^8)*t^2 + ((a^12 - 5/2*a^8 + 2*a^4 - 1/2)/a^8)*t + ((1/4*a^16 - a^12 + 3/2*a^8 - a^4 + 1/4)/a^8)*s,\n", " ((3*a^24 - 9/2*a^20 + 3/4*a^16 + a^12 - 1/8)/a^24)*t^3 + ((9/2*a^28 - 21/2*a^24 + 9/2*a^20 + 9/2*a^16 - 3*a^12 + 3/4*a^8 - 3/2*a^4 + 3/4)/a^24)*t^2 + (((3/4*a^32 - 3*a^28 + 9/2*a^24 - 3*a^20 + 3/8*a^16 + 3/2*a^12 - 9/4*a^8 + 3/2*a^4 - 3/8)/a^24)*s + (a^36 - 6*a^28 + 11/2*a^24 + 3*a^20 - 3*a^16 - a^12 - 3/2*a^8 + 3*a^4 - 1)/a^24)*t + ((1/8*a^48 - a^36 - 3/4*a^32 + 9/2*a^28 - 5/2*a^24 - 21/8*a^16 + 3/2*a^12 + 3*a^8 - 3*a^4 + 3/4)/a^24)*s,\n", " ((12*a^48 - 24*a^44 + 9*a^40 + 9/2*a^36 - 3/2*a^24 + 1/16)/a^48)*t^4 + ((24*a^52 - 66*a^48 + 93/2*a^44 + 9*a^40 - 27/2*a^36 + 9*a^32 - 18*a^28 + 9*a^24 - 3/4*a^8 + 3/2*a^4 - 3/4)/a^48)*t^3 + (((3*a^56 - 27/2*a^52 + 24*a^48 - 21*a^44 + 15/2*a^40 + 9/2*a^36 - 9*a^32 + 6*a^28 - 3/2*a^24 + 3/8*a^16 - 3/2*a^12 + 9/4*a^8 - 3/2*a^4 + 3/8)/a^48)*s + (6*a^60 + 6*a^56 - 63*a^52 + 78*a^48 - 24*a^44 + 15*a^40 - 15*a^36 - 36*a^32 + 48*a^28 - 14*a^24 - 21/4*a^16 + 3*a^12 + 15/2*a^8 - 9*a^4 + 11/4)/a^48)*t^2 + (((1/2*a^72 - 5/2*a^60 - 12*a^56 + 81/2*a^52 - 161/4*a^48 + 45/2*a^44 - 39/2*a^40 + 19/2*a^36 + 27/2*a^32 - 21*a^28 + 29/4*a^24 + 9/2*a^20 - 6*a^16 + 12*a^12 - 69/4*a^8 + 21/2*a^4 - 9/4)/a^48)*s + (a^72 - 8*a^60 - 6*a^56 + 36*a^52 - 41/2*a^48 - 21*a^40 + 16*a^36 + 27*a^32 - 42*a^28 + 16*a^24 + 21/2*a^16 - 6*a^12 - 12*a^8 + 12*a^4 - 3)/a^48)*t + ((3/16*a^32 - 3/2*a^28 + 21/4*a^24 - 21/2*a^20 + 105/8*a^16 - 21/2*a^12 + 21/4*a^8 - 3/2*a^4 + 3/16)/a^48)*s^2 + ((1/16*a^96 - 3/2*a^72 + 9/2*a^60 + 9*a^56 - 24*a^52 + 51/4*a^48 - 6*a^36 - 75/16*a^32 + 57/2*a^28 - 41/2*a^24 + 6*a^20 - 51/4*a^16 + 3*a^12 + 15*a^8 - 12*a^4 + 21/8)/a^48)*s,\n", " ((60*a^80 - 150*a^76 + 90*a^72 + 25*a^68 - 20*a^64 + 5/2*a^60 - 10*a^56 + 5/4*a^52 + a^40 + 5/16*a^32 - 1/32)/a^80)*t^5 + ((150*a^84 - 480*a^80 + 435*a^76 + 50*a^72 - 495/2*a^68 + 165*a^64 - 565/4*a^60 + 155/2*a^56 - 35/4*a^52 - 10*a^48 + 20*a^44 - 55/4*a^40 + 15/2*a^36 - 15/4*a^32 + 5/8*a^8 - 5/4*a^4 + 5/8)/a^80)*t^4 + (((15*a^88 - 75*a^84 + 150*a^80 - 295/2*a^76 + 115/2*a^72 + 135/4*a^68 - 70*a^64 + 55*a^60 - 45/2*a^56 + 15/4*a^52 + 15/8*a^48 - 15/2*a^44 + 45/4*a^40 - 15/2*a^36 + 15/8*a^32 - 5/16*a^16 + 5/4*a^12 - 15/8*a^8 + 5/4*a^4 - 5/16)/a^80)*s + (40*a^92 + 55*a^88 - 1135/2*a^84 + 830*a^80 - 235*a^76 - 585/2*a^72 + 335*a^68 - 420*a^64 + 410*a^60 - 405/2*a^56 + 35/2*a^52 + 375/4*a^48 - 105*a^44 + 145/2*a^40 - 45*a^36 + 55/4*a^32 - 5/4*a^24 + 45/8*a^16 - 15*a^8 + 15*a^4 - 35/8)/a^80)*t^3 + (((5/2*a^104 - 5/4*a^100 - 5*a^92 - 95*a^88 + 315*a^84 - 1405/4*a^80 + 535/4*a^76 + 30*a^72 - 175/2*a^68 + 345/2*a^64 - 795/4*a^60 + 355/4*a^56 + 45/4*a^52 - 475/16*a^48 + 60*a^44 - 345/4*a^40 + 50*a^36 - 105/8*a^32 + 45/4*a^28 - 35/8*a^24 - 45/4*a^20 + 345/16*a^16 - 135/4*a^12 + 285/8*a^8 - 75/4*a^4 + 15/4)/a^80)*s + (15/2*a^104 + 20*a^96 - 105*a^92 - 140*a^88 + 650*a^84 - 1065/2*a^80 + 20*a^76 + 285/2*a^72 - 260*a^68 + 465*a^64 - 545*a^60 + 400*a^56 - 10*a^52 - 945/4*a^48 + 190*a^44 - 110*a^40 + 50*a^36 - 20*a^32 + 40*a^28 - 75/2*a^24 + 45*a^20 - 255/4*a^16 + 145/2*a^8 - 55*a^4 + 25/2)/a^80)*t^2 + (((15/16*a^64 - 15/2*a^60 + 105/4*a^56 - 105/2*a^52 + 525/8*a^48 - 105/2*a^44 + 105/4*a^40 - 15/2*a^36 + 15/32*a^32 + 15/4*a^28 - 105/8*a^24 + 105/4*a^20 - 525/16*a^16 + 105/4*a^12 - 105/8*a^8 + 15/4*a^4 - 15/32)/a^80)*s^2 + ((5/16*a^128 + 5/4*a^108 - 10*a^104 + 15/4*a^100 - 645/32*a^96 + 35*a^92 + 160*a^88 - 765/2*a^84 + 1055/4*a^80 - 65/4*a^76 - 215/4*a^72 + 165/4*a^68 - 2135/16*a^64 + 935/4*a^60 - 1365/8*a^56 + 395/4*a^52 - 385/4*a^48 + 20*a^44 + 135/2*a^40 - 65*a^36 + 2755/32*a^32 - 605/4*a^28 + 1175/8*a^24 - 405/4*a^20 + 55/2*a^16 + 185/2*a^12 - 135*a^8 + 70*a^4 - 205/16)/a^80)*s + (a^120 - 10*a^104 - 20*a^96 + 60*a^92 + 95*a^88 - 240*a^84 + 239/2*a^80 - 40*a^76 - 20*a^72 + 160*a^68 - 155*a^64 + 180*a^60 - 200*a^56 - 30*a^52 + 265/2*a^48 - 36*a^40 + 20*a^36 + 10*a^32 - 80*a^28 + 80*a^24 - 90*a^20 + 105*a^16 - 90*a^8 + 60*a^4 - 12)/a^80)*t + ((5/16*a^64 - 5/4*a^60 + 15/8*a^56 - 15/4*a^52 + 135/16*a^48 + 15/4*a^44 - 105/2*a^40 + 195/2*a^36 - 1455/16*a^32 + 265/4*a^28 - 425/8*a^24 + 45/4*a^20 + 885/16*a^16 - 315/4*a^12 + 195/4*a^8 - 15*a^4 + 15/8)/a^80)*s^2 + ((1/32*a^160 - 5/16*a^128 - a^120 - 5/4*a^108 + 10*a^104 - 5/2*a^100 + 645/32*a^96 - 25*a^92 - 90*a^88 + 150*a^84 - 119/2*a^80 - 15/4*a^72 - 85/16*a^64 + 25/2*a^60 + 45/4*a^56 - 55/2*a^52 + 70*a^48 - 515/4*a^44 + 120*a^40 - 100*a^36 + 245/16*a^32 + 110*a^28 - 235/2*a^24 + 120*a^20 - 945/8*a^16 - 15/2*a^12 + 195/2*a^8 - 60*a^4 + 45/4)/a^80)*s,\n", " ((360*a^120 - 1080*a^116 + 900*a^112 + 75*a^108 - 555/2*a^104 + 30*a^100 - 55*a^96 + 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2331/4*a^236 - 31059/32*a^232 + 30387/16*a^228 - 503307/128*a^224 + 7182*a^220 - 146517/16*a^216 + 63*a^212 + 110061/4*a^208 - 1950837/32*a^204 + 3502863/32*a^200 - 2893401/16*a^196 + 6144873/64*a^192 + 11095875/32*a^188 - 55374003/64*a^184 + 29399781/32*a^180 - 65930643/128*a^176 + 153135*a^172 + 111567/8*a^168 - 3125115/16*a^164 + 20743947/128*a^160 + 10409721/32*a^156 - 36054711/64*a^152 + 7576695/32*a^148 - 36512259/128*a^144 + 11206629/16*a^140 - 3229317/8*a^136 - 2775171/8*a^132 + 125713287/256*a^128 + 226233/8*a^124 - 9685725/32*a^120 - 1350405/4*a^116 + 72126873/64*a^112 - 19198473/16*a^108 + 4422663/4*a^104 - 13765185/16*a^100 - 1768431/64*a^96 + 2651607/8*a^92 + 6184143/16*a^88 - 3693081/8*a^84 - 2880171/16*a^80 + 20670615/32*a^76 - 10721865/8*a^72 + 16217775/8*a^68 - 228075435/128*a^64 + 9113265/16*a^60 + 33532065/32*a^56 - 40084065/16*a^52 + 215728695/64*a^48 - 22369095/8*a^44 + 5018895/8*a^40 + 10571715/8*a^36 - 132085485/64*a^32 + 2373840*a^28 - 7992495/4*a^24 + 1974105/4*a^20 + 27152685/32*a^16 - 7795305/8*a^12 + 3709125/8*a^8 - 109620*a^4 + 169155/16)/a^288)*s^2 + ((1/512*a^576 - 9/256*a^512 - 9/32*a^460 + 9/16*a^456 + 9/512*a^448 - a^432 - 21/16*a^420 + 63/8*a^412 - 63/8*a^408 + 27/2*a^400 - 135/32*a^392 + 45/2*a^380 + 351/4*a^376 - 1185/4*a^372 + 189/2*a^368 + 987/4*a^360 + 126*a^356 - 92745/128*a^352 + 12573/16*a^348 - 22563/32*a^344 + 63*a^340 - 426327/64*a^336 + 14238*a^332 - 20412*a^328 + 19425*a^324 + 54810*a^320 - 619983/8*a^316 + 65520*a^312 - 1511595/4*a^308 + 24914799/64*a^304 + 9192897/16*a^300 - 5080383/4*a^296 + 1632897/2*a^292 - 23196305/128*a^288 - 693/2*a^284 + 945/8*a^280 - 3255/16*a^276 + 94815/64*a^272 - 3771*a^268 + 8685/2*a^264 - 57105/8*a^260 - 2756475/256*a^256 + 271953/8*a^252 - 483147/32*a^248 + 1465443/16*a^244 - 10625265/64*a^240 - 1061865/8*a^236 + 15261291/32*a^232 - 728421/2*a^228 + 12002139/128*a^224 - 42021/8*a^220 + 87717/16*a^216 + 23751/4*a^212 - 1009575/32*a^208 + 1526385/32*a^204 - 303471/4*a^200 + 2130831/16*a^196 + 2516367/256*a^192 - 7732305/16*a^188 + 12187791/16*a^184 - 408492*a^180 - 8171523/64*a^176 + 1806363/8*a^172 - 368091/4*a^168 + 406791*a^164 - 8704521/16*a^160 - 2286207/4*a^156 + 25566471/16*a^152 - 8459955/8*a^148 + 13532953/64*a^144 - 984249/8*a^140 + 543753/4*a^136 - 305319/2*a^132 + 49170303/128*a^128 - 1358721/8*a^124 - 7724787/16*a^120 + 761670*a^116 - 22169889/32*a^112 + 1280573/2*a^108 - 6774075/16*a^104 + 167706*a^100 - 35055279/64*a^96 + 9985311/8*a^92 - 19810665/16*a^88 + 6126057/8*a^84 - 19401291/32*a^80 + 6685245/16*a^76 + 1049825/2*a^72 - 5826555/4*a^68 + 47477745/64*a^64 + 644805*a^60 - 463995/2*a^56 - 2945565/2*a^52 + 39772845/16*a^48 - 4347945/2*a^44 + 3004155/4*a^40 + 447930*a^36 - 10286325/16*a^32 + 1252125*a^28 - 4499145/2*a^24 + 1649970*a^20 + 1023435/8*a^16 - 1938195/2*a^12 + 1272915/2*a^8 - 181440*a^4 + 80325/4)/a^288)*s]" ] }, "execution_count": 281, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# IGF of 2-CNF (we take a parameter a = sqrt{sqrt{2}})\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "cnf = dt*exp_had_prod(exp(s*cscc.subs(z=z/2)-t*scd/2),ddotset,N)\n", "CNF = [cnf[i] * i.factorial() for i in srange(disp)]\n", "CNF\n", "# For N = 20, the calculations took about 15 minutes" ] }, { "cell_type": "code", "execution_count": 282, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " t,\n", " 9*t^2 + s + 6*t,\n", " 543*t^3 + 3*(31*s + 360)*t + 774*t^2 + 1606*s,\n", " 182785*t^4 + 18*(1963*s + 46950)*t^2 + 390132*t^3 + 3*s^2 + 44*(18821*s + 37084)*t + 12864042*s,\n", " 312314367*t^5 + 30*(2086741*s + 78291942)*t^3 + 875530260*t^4 + 200*(8081847*s + 33251042)*t^2 + 16060*s^2 + 15*(511*s^2 + 2229402298*s + 1233833344)*t + 1035697286504*s,\n", " 2573579309057*t^6 + 15*(34771820545*s + 1836104171628)*t^4 + 8887318855650*t^5 + 60*(234641666875*s + 1526936187618)*t^3 + 15*s^3 + 45*(1568769*s^2 + 7003085894786*s + 8068294587808)*t^2 + 218752990*s^2 + 6*(36806965*s^2 + 2138819532488878*s + 256818479707520)*t + 1137724245192445576*s,\n", " 99110960341508095*t^7 + 63*(319904112719189*s + 22382442035882620)*t^5 + 405905598910414890*t^6 + 70*(8001505062568745*s + 73831753980680916)*t^4 + 315*(8922690901*s^2 + 40891385226706282*s + 73808042117823616)*t^3 + 168630*s^3 + 42*(258258419565*s^2 + 13296521758264855082*s + 3309832127931253568)*t^2 + 22472730817404*s^2 + 7*(122865*s^3 + 2485879069570*s^2 + 9346686403862956505048*s + 135103211268129911808)*t + 19275699325699284398997808*s,\n", " 17456249235426084913153*t^8 + 28*(126904194747370110977*s + 11304993958269225271470)*t^6 + 82650223016596904345544*t^7 + 56*(1806950471793025808811*s + 22272840374614812244840)*t^5 + 1050*(476253944728781*s^2 + 2244124383375652293582*s + 5686507641655437370280)*t^4 + 105*s^4 + 168*(13438379974869085*s^2 + 613601966443036674328178*s + 236911197744127657936272)*t^3 + 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316464797764954712848514761605271409690278754165274954386232495110595837024120*s^6 + 12268269376838420497924058013868879510582100427949111443565841000306683372858585107016011200640*s^5 + 13258260129590604607394625575350428386005117273694841307618404759369563216615041493876083410302916577129540656*s^4 + 464436987196649272388126131016696496273589236555056014650273363476588309299532849198083690149165897787428133053976167643008*s^3 + 3912728354904170701801158765943850331711719719274880807211873386887640253394971177660283159694716285650454341796095815307428574188592008917473909504256*s^2 + 611870965682955966471430199336992077907659543685068349691264975742267121878863656748051224259644814498243003937457216418045514894680552167158599835504518823783105110594923600344306546688*s + 13404628762116454488465840921098690194616333758129942834108375904608037717000453163637201809391513273664036415045905094021580460296636489453797904549412864)*t^2 + 9847222435543157805192937067357951271232569194129483500907960125903345584410703615404139640588890956754929998841730955516913935473528117264492739772835688743452594176*s^2 + 19*(4736067309212414175*s^9 + 9435425465644868705878497029862284017200*s^8 + 2134416756610460845871493498889270331150013844982235752794680*s^7 + 9754499363910701490794260792229487763923062166920541842574664914903084596454400*s^6 + 1090203223748577443589127960524293393998967643645540109166487005809574015074591899389242010949360*s^5 + 3751027358609035386535490035150372722030112639209352870740796428273122665077052066426282297498579788216521622784*s^4 + 487377637317868469526704164306698620510880029630220989462483110022547899514413856364522197282569031825640032095420530152728832*s^3 + 65644633493628998301084920103305960414241926389864009594924092401843337194281056523058427868079672656037515046310679548246985485517919870487440015857351819264*s^2 + 2335969592534341734846445989781440726319198221506541356440919148104150814855431227385734598597451996649081566078759981614417036374623501160951541094383025598349440326537890492348477070208356892672*s + 739845671839023485099057702628416878660441211325580421700063133143315424249686835913294520817428417461249560895704998389311178350371352288083817626916261724160)*t + 80263304117426476159047797543163947531906894435969580136809190481139563434236982122744001003209174414285402067740650764605666397087324831155987004502670832435475322879659307954415090718150785028137648193536*s]" ] }, "execution_count": 282, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Numbers of 2-CNF\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "CNFsub = [cnf[i].subs(a=sqrt(sqrt(2))) * 2**(i*i) * i.factorial() for i in srange(N)]\n", "CNFsub" ] }, { "cell_type": "code", "execution_count": 283, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 1,\n", " 16,\n", " 4096,\n", " 16777216,\n", " 1099511627776,\n", " 1152921504606846976,\n", " 19342813113834066795298816,\n", " 5192296858534827628530496329220096,\n", " 22300745198530623141535718272648361505980416,\n", " 1532495540865888858358347027150309183618739122183602176,\n", " 1684996666696914987166688442938726917102321526408785780068975640576,\n", " 29642774844752946028434172162224104410437116074403984394101141506025761187823616,\n", " 8343699359066055009355553539724812947666814540455674882605631280555545803830627148527195652096,\n", " 37576681324381331646231689548629392438010920782533117931316655544515344401833735095419183974156299248510959616,\n", " 2707685248164858261307045101702230179137145581421695874189921465443966120903931272499975005961073806735733604454495675614232576,\n", " 3121748550315992231381597229793166305748598142664971150859156959625371738819765620120306103063491971159826931121406622895447975679288285306290176,\n", " 57586096570152913699974892898380567793532123114264532903689671329431521032595044740083720782129802971518987656109067457577065805510327036019308994315074097345724416,\n", " 16996415770136547158066822609678996074546979767265021542382212422412913915547271767653200072487337141404458543559888032491090538804886631661104639320530795262202600666732583009015300096,\n", " 80263304161809898486953580976564463280492245526476651908848280381297792881730359224146523075524726123458602430056430323990164676669064390001339947061948865508349970567755807467524166227482951618519489314816]" ] }, "execution_count": 283, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Verification, all 2-CNF\n", "CNFsubOneTS = [CNFsub[i].subs(s=1,t=1) for i in srange(N)]\n", "CNFsubOneTS" ] }, { "cell_type": "code", "execution_count": 284, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 16,\n", " 256,\n", " 4096,\n", " 65536,\n", " 1048576,\n", " 16777216,\n", " 268435456,\n", " 4294967296,\n", " 68719476736,\n", " 1099511627776,\n", " 17592186044416,\n", " 281474976710656,\n", " 4503599627370496,\n", " 72057594037927936,\n", " 1152921504606846976,\n", " 18446744073709551616,\n", " 295147905179352825856,\n", " 4722366482869645213696]" ] }, "execution_count": 284, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Frac = [CNFsubOneTS[i+1]/CNFsubOneTS[i] for i in srange(N-1)]\n", "Frac" ] }, { "cell_type": "code", "execution_count": 285, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16]" ] }, "execution_count": 285, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Frac2 = [Frac[i+1]/Frac[i] for i in srange(N-2)]\n", "Frac2" ] }, { "cell_type": "code", "execution_count": 286, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " 0,\n", " 1,\n", " 1606,\n", " 12864042,\n", " 1035697286504,\n", " 1137724245192445576,\n", " 19275699325699284398997808,\n", " 5187221831835207014808821762704464,\n", " 22294619997952835408033533526860039478469248,\n", " 1532378620903971051220118091442016459229955421936318080,\n", " 1684961314076455535365379591999039797953859605423390876101655503104,\n", " 29642605227454854630551408130507373873482741310983797918043445583780311814302464,\n", " 8343686428662293190981251306447000897822543196781971769510910299825116404717213746936206525440,\n", " 37576665646168253166866482504052164868974270211375861912370715699343527242379360781177046860802446211634115584,\n", " 2707684945557451236582206955141024326876877071965141524178301505534293508529886661571956358105100236930563137233767989679806464,\n", " 3121748457280635564865428301743914485519056577281653558208335774024045241858742454148748399032510665780515208957477600857184656604559187452143616,\n", " 57586096114287444134995809173660516831168083903214752826003829800109384494951683651682173505345824514486845196932414284209416836541778724905135505112784920768053248,\n", " 16996415734520969346976316634185422653151718716397360127143532564908212078333732597468381359902999464657535664852571684255507969168905044622778371903233605242287145369230273632552517632,\n", " 80263304117426476159047797543163947531906894435969580136809190481139563434236982122744001003209174414285402067740650764605666397087324831155987004502670832435475322879659307954415090718150785028137648193536]" ] }, "execution_count": 286, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Verification, contradictory strongly connected components\n", "CNFsubZeroT = [CNFsub[i].diff(s).subs(s=0,t=0) for i in srange(N)]\n", "CNFsubZeroT" ] }, { "cell_type": "code", "execution_count": 287, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]" ] }, "execution_count": 287, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ZeroT = [CNFsubZeroT[i] - CSCCsub[i] for i in srange(N)]\n", "ZeroT" ] }, { "cell_type": "code", "execution_count": 288, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 1,\n", " 15,\n", " 2397,\n", " 3049713,\n", " 28694311447,\n", " 2034602766692687,\n", " 1115068294703296663717,\n", " 4795802950152171162502013473,\n", " 163220487110350216972297148097903343,\n", " 44164928411665942510460654486808636655906527,\n", " 95265878966579205334990504387281268292082212633510733,\n", " 1640680461860732121820130368901670064509937267196250426559284369,\n", " 225797401059395491388226781486494859477336476383420624645574889279833291655,\n", " 248449096473444603309693803281786498777604826860716062491779495426821364225621671245231,\n", " 2186249373678136497929092867348607033501533753383379007905060966599312858378097566945226476160367797,\n", " 153876507870619064316245403952108076113373975727815029772783646372965670866193991271419148214709969273888724439361,\n", " 86634689646631695792362052797375058873054204228562227682034608063501723861570442427435886561302900664407051568560067003936816351,\n", " 390191774172218805142782958749241933165131883043872896011181628217746441735276665013850091802123482655700143733528694386779753133685046362583743,\n", " 14058596106872313778290075591271893305303313138868553609649111587958343503927693511047338576434109913612519458536994817454174322323926252697999933306146761085405]" ] }, "execution_count": 288, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Verification, satisfiable 2-CNF\n", "CNFsubZeroS = [CNFsub[i].subs(s=0,t=1) for i in srange(N)]\n", "CNFsubZeroS" ] }, { "cell_type": "code", "execution_count": 289, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]" ] }, "execution_count": 289, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ZeroS = [CNFsubZeroS[i] - SATsub[i] for i in srange(N)]\n", "ZeroS" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 290, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 0,\n", " 0,\n", " 0,\n", " (-12*a^10*t + 12*a^10)*w,\n", " 0,\n", " 0,\n", " 0,\n", " (5040*a^20*t^2 + (-10080*a^28 + 20160*a^24 - 20160*a^20)*t + (20160*a^28 - 80640*a^24 + 120960*a^20 - 80640*a^16 + 20160*a^12)*s - 10080*a^28 + 60480*a^24 - 105840*a^20 + 80640*a^16 - 20160*a^12)*w^2,\n", " 0]" ] }, "execution_count": 290, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Asymptotics of 2-CNF, part 1 (we take a parameter a = sqrt{sqrt{2}})\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "part1 = exp((1-t)*scd.subs(z = a^(10)*z^4*w)/2 + (s-1)*cscc.subs(z = a^(6)*z^4*w))\n", "PART1 = [part1[i] * i.factorial() for i in srange(disp)]\n", "PART1" ] }, { "cell_type": "code", "execution_count": 291, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " 1,\n", " 0,\n", " -6*a^9*w,\n", " 0,\n", " (-60*a^26 + 120*a^22)*w^2 + (-60*a^10*t + 60*a^10)*w,\n", " 0,\n", " (-840*a^51 + 5040*a^43 - 5040*a^39)*w^3 + (2520*a^19*t - 2520*a^19)*w^2,\n", " 0,\n", " (-15120*a^84 + 120960*a^72 + 90720*a^68 - 544320*a^64 + 362880*a^60)*w^4 + ((90720*a^36 - 181440*a^32)*t - 90720*a^36 + 181440*a^32)*w^3 + (45360*a^20*t^2 + (-90720*a^28 + 181440*a^24 - 181440*a^20)*t + (181440*a^28 - 725760*a^24 + 1088640*a^20 - 725760*a^16 + 181440*a^12)*s - 90720*a^28 + 544320*a^24 - 952560*a^20 + 725760*a^16 - 181440*a^12)*w^2]" ] }, "execution_count": 291, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Asymptotics of 2-CNF, part 2 (we take a parameter a = sqrt{sqrt{2}})\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "part2 = z * part1 * exp_had_prod(PhiTwoTwoFour,1/g.subs(z = a^8*z^2*w),N)\n", "PART2 = [part2[i] * i.factorial() for i in srange(disp)]\n", "PART2" ] }, { "cell_type": "code", "execution_count": 292, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " s,\n", " 0,\n", " (6*a^6*s*t - 6*a^6*s)*w,\n", " 0,\n", " (((120*a^12 - 60*a^8)*s)*t^2 + ((60*a^16 - 240*a^12 + 60*a^8)*s)*t + (-60*a^16 + 120*a^12)*s)*w^2 + (-60*s*t + 60*s)*w,\n", " 0,\n", " (((5040*a^18 - 5040*a^14 + 840*a^6)*s)*t^3 + ((5040*a^22 - 15120*a^18 + 5040*a^14 + 5040*a^10 - 2520*a^6)*s)*t^2 + ((840*a^30 - 10080*a^22 + 15120*a^18 - 5040*a^10 + 1680*a^6)*s)*t + (-840*a^30 + 5040*a^22 - 5040*a^18)*s)*w^3 + (-2520*a^6*s*t^2 + (((2520*a^8 + 2520)/a^2)*s)*t - 2520/a^2*s)*w^2,\n", " 0,\n", " (((362880*a^24 - 544320*a^20 + 90720*a^16 + 120960*a^12 - 15120)*s)*t^4 + ((544320*a^28 - 1632960*a^24 + 907200*a^20 + 544320*a^16 - 423360*a^12 + 90720*a^8 - 181440*a^4 + 90720)*s)*t^3 + ((120960*a^36 + 90720*a^32 - 1632960*a^28 + 2388960*a^24 - 362880*a^20 - 680400*a^16 + 241920*a^12 - 453600*a^8 + 544320*a^4 - 166320)*s)*t^2 + ((15120*a^48 - 241920*a^36 - 181440*a^32 + 1632960*a^28 - 1481760*a^24 + 45360*a^16 + 60480*a^12 + 362880*a^8 - 362880*a^4 + 90720)*s)*t + (-15120*a^48 + 120960*a^36 + 90720*a^32 - 544320*a^28 + 362880*a^24)*s)*w^4 + (((-181440*a^12 + 90720*a^8)*s)*t^3 + ((-90720*a^16 + 362880*a^12 - 181440*a^8 + 181440*a^4)*s)*t^2 + (((90720*a^20 - 181440*a^16 + 90720*a^12 - 181440*a^8 + 90720*a^4 - 181440)/a^4)*s)*t + ((-90720*a^4 + 181440)/a^4)*s)*w^3 + (45360/a^16*s*t^2 + (((-90720*a^8 + 181440*a^4 - 181440)/a^16)*s)*t + ((181440*a^16 - 725760*a^12 + 1088640*a^8 - 725760*a^4 + 181440)/a^24)*s^2 + ((-90720*a^16 + 544320*a^12 - 952560*a^8 + 725760*a^4 - 181440)/a^24)*s)*w^2,\n", " 0,\n", " ((((39916800*a^40 - 79833600*a^36 + 29937600*a^32 + 19958400*a^28 - 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2067966706950144000*a^12 - 144453556735488000*a^8 + 669048052248576000*a^4 - 182467650613248000)/a^138)*s^2 + ((5068545850368000*a^96 - 121645100408832000*a^72 - 2534272925184000*a^64 + 364935301226496000*a^60 + 729870602452992000*a^56 - 1915910331439104000*a^52 + 1049188991026176000*a^48 - 91233825306624000*a^44 - 35479820952576000*a^40 - 547402951839744000*a^36 + 64623959592192000*a^32 + 2093309436201984000*a^28 - 1955191561779456000*a^24 + 904735434290688000*a^20 - 1736293737866688000*a^16 + 1216451004088320000*a^12 + 570211408166400000*a^8 - 790693152657408000*a^4 + 197673288164352000)/a^138)*s)*w^5]" ] }, "execution_count": 292, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# CGF of 2-CNF (we take a parameter a = sqrt{sqrt{2}})\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "qcnf = s * dt.subs(z = a^6*z^2*w) * exp_had_prod(PhiTwoFourTwo,part2,N)\n", "QCNF = [qcnf[i] * i.factorial() for i in srange(N)]\n", "QCNF" ] }, { "cell_type": "code", "execution_count": 293, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " s,\n", " 0,\n", " (a^12*s*t - a^12*s)*w,\n", " 0,\n", " (((a^32 - 1/2*a^28)*s)*t^2 + ((1/2*a^36 - 2*a^32 + 1/2*a^28)*s)*t + (-1/2*a^36 + a^32)*s)*w^2 + (-1/2*a^20*s*t + 1/2*a^20*s)*w,\n", " 0,\n", " (((a^60 - a^56 + 1/6*a^48)*s)*t^3 + ((a^64 - 3*a^60 + a^56 + a^52 - 1/2*a^48)*s)*t^2 + ((1/6*a^72 - 2*a^64 + 3*a^60 - a^52 + 1/3*a^48)*s)*t + (-1/6*a^72 + a^64 - a^60)*s)*w^3 + (-1/2*a^48*s*t^2 + ((1/2*a^48 + 1/2*a^40)*s)*t - 1/2*a^40*s)*w^2,\n", " 0,\n", " (((a^96 - 3/2*a^92 + 1/4*a^88 + 1/3*a^84 - 1/24*a^72)*s)*t^4 + ((3/2*a^100 - 9/2*a^96 + 5/2*a^92 + 3/2*a^88 - 7/6*a^84 + 1/4*a^80 - 1/2*a^76 + 1/4*a^72)*s)*t^3 + ((1/3*a^108 + 1/4*a^104 - 9/2*a^100 + 79/12*a^96 - a^92 - 15/8*a^88 + 2/3*a^84 - 5/4*a^80 + 3/2*a^76 - 11/24*a^72)*s)*t^2 + ((1/24*a^120 - 2/3*a^108 - 1/2*a^104 + 9/2*a^100 - 49/12*a^96 + 1/8*a^88 + 1/6*a^84 + a^80 - a^76 + 1/4*a^72)*s)*t + (-1/24*a^120 + 1/3*a^108 + 1/4*a^104 - 3/2*a^100 + a^96)*s)*w^4 + (((-1/2*a^84 + 1/4*a^80)*s)*t^3 + ((-1/4*a^88 + a^84 - 1/2*a^80 + 1/2*a^76)*s)*t^2 + ((1/4*a^88 - 1/2*a^84 + 1/4*a^80 - 1/2*a^76 + 1/4*a^72 - 1/2*a^68)*s)*t + (-1/4*a^72 + 1/2*a^68)*s)*w^3 + (1/8*a^56*s*t^2 + ((-1/4*a^64 + 1/2*a^60 - 1/2*a^56)*s)*t + (1/2*a^64 - 2*a^60 + 3*a^56 - 2*a^52 + 1/2*a^48)*s^2 + (-1/4*a^64 + 3/2*a^60 - 21/8*a^56 + 2*a^52 - 1/2*a^48)*s)*w^2,\n", " 0,\n", " (((a^140 - 2*a^136 + 3/4*a^132 + 1/2*a^128 - 1/6*a^124 - 1/12*a^116 + 1/120*a^100)*s)*t^5 + ((2*a^144 - 13/2*a^140 + 5*a^136 + 23/12*a^132 - 19/6*a^128 + 7/6*a^124 - a^120 + 13/24*a^116 - 1/12*a^108 + 1/6*a^104 - 1/12*a^100)*s)*t^4 + ((1/2*a^152 + 7/12*a^148 - 8*a^144 + 38/3*a^140 - 3*a^136 - 67/12*a^132 + 25/6*a^128 - 7/2*a^124 + 7/2*a^120 - 37/24*a^116 + a^108 - a^104 + 7/24*a^100)*s)*t^3 + ((1/12*a^164 + 1/6*a^156 - 3/2*a^152 - 49/24*a^148 + 12*a^144 - 45/4*a^140 - 1/6*a^136 + 27/8*a^132 - 8/3*a^128 + 29/6*a^124 - 5*a^120 + 37/12*a^116 - 29/12*a^108 + 11/6*a^104 - 5/12*a^100)*s)*t^2 + ((1/120*a^180 - 1/6*a^164 - 1/3*a^156 + 3/2*a^152 + 53/24*a^148 - 8*a^144 + 61/12*a^140 + 1/6*a^136 - 11/24*a^132 + 7/6*a^128 - 7/3*a^124 + 5/2*a^120 - 2*a^116 + 3/2*a^108 - a^104 + 1/5*a^100)*s)*t + (-1/120*a^180 + 1/12*a^164 + 1/6*a^156 - 1/2*a^152 - 3/4*a^148 + 2*a^144 - a^140)*s)*w^5 + (((-1/2*a^128 + 1/2*a^124 - 1/12*a^116)*s)*t^4 + ((-1/2*a^132 + 3/2*a^128 - 3/4*a^124 + 1/12*a^116)*s)*t^3 + ((-1/12*a^140 + a^132 - 3/2*a^128 + 1/4*a^124 + 1/3*a^116 - 1/2*a^112)*s)*t^2 + ((1/12*a^140 - 1/2*a^132 + 1/2*a^128 - 1/4*a^116 + 1/2*a^112 - 1/2*a^108 + 1/2*a^104)*s)*t + (-1/12*a^116 + 1/2*a^108 - 1/2*a^104)*s)*w^4 + (1/8*a^100*s*t^3 + ((-1/4*a^108 + 1/2*a^104 - 1/2*a^100 - 1/8*a^84)*s)*t^2 + ((1/2*a^108 - 2*a^104 + 3*a^100 - 2*a^96 + 1/2*a^92)*s^2 + (-1/4*a^108 + 3/2*a^104 - 21/8*a^100 + 2*a^96 - 1/4*a^92 - 1/2*a^88 + 1/2*a^84)*s)*t + (-1/2*a^92 + 2*a^88 - 3*a^84 + 2*a^80 - 1/2*a^76)*s^2 + (1/4*a^92 - 3/2*a^88 + 21/8*a^84 - 2*a^80 + 1/2*a^76)*s)*w^3,\n", " 0,\n", " (((a^192 - 5/2*a^188 + 3/2*a^184 + 13/24*a^180 - 1/2*a^176 - 7/72*a^168 + 1/24*a^164 + 1/60*a^152 - 1/720*a^132)*s)*t^6 + ((5/2*a^196 - 9*a^192 + 71/8*a^188 + 3/2*a^184 - 49/8*a^180 + 11/4*a^176 - 35/24*a^172 + 5/4*a^168 - 7/24*a^164 - 1/6*a^160 + 1/3*a^156 - 7/40*a^152 + 1/48*a^140 - 1/24*a^136 + 1/48*a^132)*s)*t^5 + ((2/3*a^204 + a^200 - 103/8*a^196 + 821/36*a^192 - 197/24*a^188 - 265/24*a^184 + 31/3*a^180 - 73/12*a^176 + 111/16*a^172 - 173/36*a^168 + 17/24*a^164 + 25/12*a^160 - 79/36*a^156 + 2/3*a^152 + 5/48*a^148 + 1/12*a^144 - 11/24*a^140 + 5/12*a^136 - 17/144*a^132)*s)*t^4 + ((1/8*a^216 - 1/24*a^212 + 1/2*a^208 - 61/24*a^204 - 61/12*a^200 + 625/24*a^196 - 241/9*a^192 + 1/3*a^188 + 95/8*a^184 - 125/16*a^180 + 125/12*a^176 - 41/3*a^172 + 227/24*a^168 - 3/4*a^164 - 21/4*a^160 + 185/48*a^156 + 1/4*a^152 - 11/8*a^148 - 2/3*a^144 + 107/48*a^140 - 35/24*a^136 + 5/16*a^132)*s)*t^3 + ((1/60*a^232 + 1/24*a^220 - 25/72*a^216 + 13/240*a^212 - 3/2*a^208 + 29/8*a^204 + 205/24*a^200 - 619/24*a^196 + 154/9*a^192 + 15/8*a^188 - 14/3*a^184 + 565/144*a^180 - 29/4*a^176 + 469/48*a^172 - 311/36*a^168 + 15/8*a^164 + 11/3*a^160 - 245/144*a^156 - 397/120*a^152 + 163/48*a^148 + 17/12*a^144 - 91/24*a^140 + 25/12*a^136 - 137/360*a^132)*s)*t^2 + ((1/720*a^252 - 1/30*a^232 - 1/12*a^220 + 23/72*a^216 - 1/80*a^212 + 3/2*a^208 - 55/24*a^204 - 143/24*a^200 + 101/8*a^196 - 221/36*a^192 - 3/8*a^188 + 5/6*a^184 - 31/36*a^180 + 2/3*a^176 - 19/12*a^172 + 17/6*a^168 - 19/12*a^164 - 1/3*a^160 - 7/24*a^156 + 51/20*a^152 - 17/8*a^148 - 5/6*a^144 + 2*a^140 - a^136 + 1/6*a^132)*s)*t + (-1/720*a^252 + 1/60*a^232 + 1/24*a^220 - 7/72*a^216 - 1/2*a^208 + 13/24*a^204 + 3/2*a^200 - 5/2*a^196 + a^192)*s)*w^6 + (((-1/2*a^180 + 3/4*a^176 - 1/8*a^172 - 1/6*a^168 + 1/48*a^156)*s)*t^5 + ((-3/4*a^184 + 9/4*a^180 - 3/2*a^176 - 1/8*a^172 + 1/6*a^168 - 1/8*a^164 + 1/3*a^160 - 7/48*a^156)*s)*t^4 + ((-1/6*a^192 - 1/8*a^188 + 9/4*a^184 - 41/12*a^180 + 5/4*a^176 - 5/16*a^172 + 5/12*a^168 + 5/8*a^164 - 13/12*a^160 + 17/48*a^156)*s)*t^3 + ((-1/48*a^204 + 1/3*a^192 + 1/4*a^188 - 13/6*a^184 + 25/12*a^180 - a^176 + 13/8*a^172 - 5/6*a^168 - 7/8*a^164 + 2/3*a^160 + 7/48*a^156)*s)*t^2 + ((1/48*a^204 - 1/6*a^192 - 1/8*a^188 + 2/3*a^184 - 5/12*a^180 + 1/2*a^176 - 25/24*a^172 + 5/12*a^168 + 3/8*a^164 - 1/12*a^160 - 1/2*a^156 + 3/4*a^152 - 1/2*a^148)*s)*t + (-1/48*a^172 + 1/6*a^160 + 1/8*a^156 - 3/4*a^152 + 1/2*a^148)*s)*w^5 + (((1/8*a^152 - 1/16*a^148)*s)*t^4 + ((-1/4*a^160 + 11/16*a^156 - 7/8*a^152 + 5/16*a^148 - 1/8*a^136)*s)*t^3 + ((1/2*a^160 - 9/4*a^156 + 4*a^152 - 7/2*a^148 + 3/2*a^144 - 1/4*a^140)*s^2 + (-1/8*a^164 + 1/4*a^160 + 3/4*a^156 - 21/8*a^152 + 49/16*a^148 - 5/4*a^144 - 1/4*a^140 + 1/2*a^136 - 1/16*a^124 + 1/8*a^120)*s)*t^2 + ((1/4*a^164 - 3/2*a^160 + 15/4*a^156 - 5*a^152 + 15/4*a^148 - 2*a^144 + 9/4*a^140 - 3*a^136 + 2*a^132 - 1/2*a^128)*s^2 + (-1/8*a^164 + a^160 - 47/16*a^156 + 35/8*a^152 - 57/16*a^148 + 7/4*a^144 - 7/4*a^140 + 21/8*a^136 - 15/8*a^132 + 3/4*a^124 - 1/2*a^120)*s)*t + (-1/4*a^132 + 3/2*a^128 - 7/2*a^124 + 4*a^120 - 9/4*a^116 + 1/2*a^112)*s^2 + (1/8*a^132 - a^128 + 45/16*a^124 - 29/8*a^120 + 9/4*a^116 - 1/2*a^112)*s)*w^4 + (-1/48*a^108*s*t^3 + ((1/8*a^116 - 1/4*a^112 + 3/16*a^108)*s)*t^2 + ((-1/4*a^116 + a^112 - 3/2*a^108 + a^104 - 1/4*a^100)*s^2 + (-1/12*a^132 + 1/2*a^124 - 1/2*a^120 - 1/4*a^116 + 49/48*a^108 - a^104 + 1/4*a^100)*s)*t + (1/6*a^144 - 4/3*a^132 - a^128 + 6*a^124 - 10/3*a^120 + 1/4*a^116 - 9/2*a^112 + 7/2*a^108 + 3*a^104 - 15/4*a^100 + a^96)*s^2 + (-1/6*a^144 + 17/12*a^132 + a^128 - 13/2*a^124 + 23/6*a^120 + 1/8*a^116 + 15/4*a^112 - 51/16*a^108 - 3*a^104 + 15/4*a^100 - a^96)*s)*w^3,\n", " 0,\n", " (((a^252 - 3*a^248 + 5/2*a^244 + 1/3*a^240 - a^236 + 1/8*a^232 - 1/12*a^228 + 1/8*a^224 - 1/72*a^216 + 1/40*a^212 - 1/120*a^208 - 1/360*a^192 + 1/5040*a^168)*s)*t^7 + ((3*a^256 - 12*a^252 + 29/2*a^248 - 1/2*a^244 - 61/6*a^240 + 25/4*a^236 - 5/2*a^232 + 169/72*a^228 - 19/24*a^224 - 1/2*a^220 + 43/60*a^216 - 9/20*a^212 + 11/120*a^208 + 1/24*a^200 - 1/12*a^196 + 31/720*a^192 - 1/240*a^176 + 1/120*a^172 - 1/240*a^168)*s)*t^6 + ((5/6*a^264 + 3/2*a^260 - 155/8*a^256 + 77/2*a^252 - 153/8*a^248 - 75/4*a^244 + 433/18*a^240 - 53/4*a^236 + 155/12*a^232 - 79/8*a^228 + 3/4*a^224 + 61/12*a^220 - 3833/720*a^216 + 143/60*a^212 - 1/6*a^208 + 1/6*a^204 - 15/16*a^200 + 7/8*a^196 - 1/4*a^192 - 1/48*a^184 - 1/24*a^180 + 7/48*a^176 - 1/8*a^172 + 5/144*a^168)*s)*t^5 + ((1/6*a^276 - 1/8*a^272 + a^268 - 281/72*a^264 - 79/8*a^260 + 199/4*a^256 - 1027/18*a^252 + 11/3*a^248 + 793/24*a^244 - 1259/48*a^240 + 53/2*a^236 - 389/12*a^232 + 1405/72*a^228 + 25/12*a^224 - 89/6*a^220 + 601/48*a^216 - 31/12*a^212 - 103/48*a^208 - 49/36*a^204 + 59/12*a^200 - 7/2*a^196 + 77/72*a^192 - 5/8*a^188 + 11/24*a^184 + 7/12*a^180 - 19/16*a^176 + 17/24*a^172 - 7/48*a^168)*s)*t^4 + ((1/40*a^292 - 1/120*a^288 + 1/8*a^280 - 7/12*a^276 + 43/120*a^272 - 47/12*a^268 + 325/48*a^264 + 545/24*a^260 - 196/3*a^256 + 841/18*a^252 + 607/80*a^248 - 181/8*a^244 + 339/16*a^240 - 135/4*a^236 + 1807/48*a^232 - 533/24*a^228 - 91/48*a^224 + 409/24*a^220 - 391/36*a^216 - 27/5*a^212 + 649/80*a^208 + 17/4*a^204 - 545/48*a^200 + 29/4*a^196 - 667/180*a^192 + 17/4*a^188 - 107/48*a^184 - 59/24*a^180 + 185/48*a^176 - 15/8*a^172 + 29/90*a^168)*s)*t^3 + ((1/360*a^312 + 1/120*a^296 - 3/40*a^292 + 1/40*a^288 - 3/8*a^280 + 3/4*a^276 - 119/240*a^272 + 707/120*a^268 - 50/9*a^264 - 583/24*a^260 + 2831/60*a^256 - 1397/60*a^252 - 163/40*a^248 + 85/12*a^244 - 1525/144*a^240 + 223/12*a^236 - 991/48*a^232 + 989/72*a^228 - 13/16*a^224 - 55/8*a^220 - 3/10*a^216 + 1639/120*a^212 - 349/30*a^208 - 85/18*a^204 + 149/12*a^200 - 187/24*a^196 + 1493/240*a^192 - 67/8*a^188 + 91/24*a^184 + 47/12*a^180 - 637/120*a^176 + 137/60*a^172 - 7/20*a^168)*s)*t^2 + ((1/5040*a^336 - 1/180*a^312 - 1/60*a^296 + 3/40*a^292 - 11/360*a^288 + 3/8*a^280 - 5/12*a^276 + 31/80*a^272 - 159/40*a^268 + 35/16*a^264 + 299/24*a^260 - 729/40*a^256 + 427/60*a^252 + 107/240*a^248 - 3/4*a^244 + 203/144*a^240 - 10/3*a^236 + 39/8*a^232 - 247/72*a^228 + 13/24*a^224 + 1/12*a^220 + 587/180*a^216 - 229/30*a^212 + 23/4*a^208 + 5/3*a^204 - 61/12*a^200 + 13/4*a^196 - 27/8*a^192 + 19/4*a^188 - 2*a^184 - 2*a^180 + 5/2*a^176 - a^172 + 1/7*a^168)*s)*t + (-1/5040*a^336 + 1/360*a^312 + 1/120*a^296 - 1/40*a^292 + 1/72*a^288 - 1/8*a^280 + 1/12*a^276 - 1/8*a^272 + a^268 - 1/3*a^264 - 5/2*a^260 + 3*a^256 - a^252)*s)*w^7 + (((-1/2*a^240 + a^236 - 3/8*a^232 - 1/4*a^228 + 1/12*a^224 + 1/24*a^216 - 1/240*a^200)*s)*t^6 + ((-a^244 + 13/4*a^240 - 11/4*a^236 - 5/24*a^232 + 11/12*a^228 - 13/24*a^224 + 2/3*a^220 - 7/24*a^216 + 1/48*a^208 - 1/12*a^204 + 11/240*a^200)*s)*t^5 + ((-1/4*a^252 - 7/24*a^248 + 4*a^244 - 79/12*a^240 + 11/4*a^236 + 5/6*a^232 - 7/6*a^228 + 25/12*a^224 - 13/6*a^220 + 17/16*a^216 - 1/3*a^212 - 19/48*a^208 + 7/12*a^204 - 3/16*a^200)*s)*t^4 + ((-1/24*a^264 - 1/12*a^256 + 3/4*a^252 + 47/48*a^248 - 35/6*a^244 + 145/24*a^240 - 13/6*a^236 + 25/24*a^232 + 7/12*a^228 - 133/48*a^224 + 17/6*a^220 - 21/8*a^216 + 13/12*a^212 + 65/48*a^208 - 17/12*a^204 + 17/48*a^200)*s)*t^3 + ((-1/240*a^280 + 1/12*a^264 + 3/16*a^256 - 3/4*a^252 - 13/12*a^248 + 11/3*a^244 - 67/24*a^240 + 2*a^236 - 29/16*a^232 - 1/12*a^228 + 19/24*a^224 - 7/4*a^220 + 139/48*a^216 - 5/12*a^212 - 101/48*a^208 + 17/12*a^204 - 37/120*a^200)*s)*t^2 + ((1/240*a^280 - 1/24*a^264 - 5/48*a^256 + 1/4*a^252 + 19/48*a^248 - 5/6*a^244 + 47/80*a^240 - 5/6*a^236 + 25/48*a^232 + 5/16*a^224 + 5/12*a^220 - 7/6*a^216 - 1/12*a^212 + 3/2*a^208 - 3/2*a^204 + 3/5*a^200)*s)*t + (-1/240*a^240 + 1/24*a^224 + 1/12*a^216 - 1/4*a^212 - 3/8*a^208 + a^204 - 1/2*a^200)*s)*w^6 + (((1/8*a^212 - 1/8*a^208 + 1/48*a^200)*s)*t^5 + ((-1/4*a^220 + 7/8*a^216 - 5/4*a^212 + 25/48*a^208 + 5/24*a^204 - 7/48*a^200 - 1/8*a^196 + 1/16*a^192)*s)*t^4 + ((1/2*a^220 - 5/2*a^216 + 5*a^212 - 59/12*a^208 + 13/6*a^204 - 1/3*a^196 + 1/12*a^192)*s^2 + (-11/48*a^224 + 3/4*a^220 - 3*a^212 + 241/48*a^208 - 23/8*a^204 - 1/3*a^200 + 29/24*a^196 - 19/48*a^192 - 1/16*a^184 + 1/8*a^180)*s)*t^3 + ((1/2*a^224 - 3*a^220 + 29/4*a^216 - 17/2*a^212 + 15/4*a^208 + 3/2*a^204 - a^200 - 5/2*a^196 + 13/4*a^192 - 3/2*a^188 + 1/4*a^184)*s^2 + (-1/24*a^232 + 1/12*a^228 - 1/12*a^224 + 5/4*a^220 - 39/8*a^216 + 31/4*a^212 - 235/48*a^208 - 23/24*a^204 + 103/48*a^200 + 9/8*a^196 - 43/16*a^192 + a^188 + 1/2*a^184 - 1/2*a^180 - 1/48*a^176 + 1/8*a^168 - 1/8*a^164)*s)*t^2 + ((1/12*a^232 - 1/3*a^228 + 13/6*a^220 - 14/3*a^216 + 7/2*a^212 + 11/12*a^208 - 8/3*a^204 - 1/2*a^200 + 23/6*a^196 - 23/6*a^192 + 3*a^188 - 15/4*a^184 + 4*a^180 - 9/4*a^176 + 1/2*a^172)*s^2 + (-1/24*a^232 + 1/4*a^228 - 3/16*a^224 - 17/12*a^220 + 47/12*a^216 - 29/8*a^212 - 13/48*a^208 + 21/8*a^204 - 3/16*a^200 - 77/24*a^196 + 169/48*a^192 - 5/2*a^188 + 149/48*a^184 - 89/24*a^180 + 25/12*a^176 + 1/4*a^172 - a^168 + 1/2*a^164)*s)*t + (-1/12*a^184 + 1/3*a^180 - 13/6*a^172 + 59/12*a^168 - 5*a^164 + 5/2*a^160 - 1/2*a^156)*s^2 + 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1/15*a^304 - 7/360*a^300)*s)*t^6 + ((-1/12*a^408 + 1/16*a^404 - 1/2*a^400 + 281/144*a^396 + 77/16*a^392 - 395/16*a^388 + 2203/72*a^384 - 115/12*a^380 - 463/72*a^376 + 145/16*a^372 - 261/16*a^368 + 131/6*a^364 - 1129/72*a^360 + 11/3*a^356 + 737/144*a^352 - 911/144*a^348 + 137/48*a^344 - 35/144*a^340 + 167/144*a^336 - 329/120*a^332 + 491/240*a^328 - 239/288*a^324 + 13/24*a^320 - 89/288*a^316 - 5/16*a^312 + 2/3*a^308 - 5/12*a^304 + 13/144*a^300)*s)*t^5 + ((-1/80*a^424 + 1/240*a^420 - 1/16*a^412 + 7/24*a^408 - 1/5*a^404 + 97/48*a^400 - 31/9*a^396 - 65/6*a^392 + 383/12*a^388 - 983/36*a^384 + 4129/480*a^380 - 253/144*a^376 - 433/72*a^372 + 20*a^368 - 1883/72*a^364 + 2915/144*a^360 - 249/32*a^356 - 65/24*a^352 + 1471/288*a^348 - 9/80*a^344 - 1553/720*a^340 - 113/36*a^336 + 323/48*a^332 - 5*a^328 + 5063/1440*a^324 - 27/8*a^320 + 449/288*a^316 + 73/48*a^312 - 121/48*a^308 + 31/24*a^304 - 337/1440*a^300)*s)*t^4 + ((-1/720*a^444 - 1/240*a^428 + 3/80*a^424 - 7/480*a^420 + 1/120*a^416 + 3/16*a^412 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+ ((1/10080*a^468 - 1/720*a^444 - 1/1440*a^436 - 1/240*a^428 + 1/80*a^424 - 1/160*a^420 + 1/120*a^416 + 61/1008*a^412 - 3/80*a^408 + 1/12*a^404 - 401/720*a^400 + 43/240*a^396 + 23/24*a^392 - 1717/1440*a^388 + 79/72*a^384 - 89/80*a^380 + 493/360*a^376 - 637/480*a^372 + 53/240*a^368 - 9/32*a^364 + 13/9*a^360 - 11/8*a^356 - 79/144*a^352 + 265/72*a^348 - 9/2*a^344 + 37/24*a^340 + 5/3*a^336 - 607/240*a^332 + 257/120*a^328 - 125/48*a^324 + 23/8*a^320 - 13/12*a^316 - a^312 + 5/4*a^308 - 1/2*a^304 + 1/14*a^300)*s)*t + (-1/10080*a^412 + 1/720*a^388 + 1/240*a^372 - 1/80*a^368 + 1/144*a^364 - 1/16*a^356 + 1/24*a^352 - 1/16*a^348 + 1/2*a^344 - 1/6*a^340 - 5/4*a^336 + 3/2*a^332 - 1/2*a^328)*s)*w^8 + (((1/8*a^356 - 1/4*a^352 + 3/32*a^348 + 1/16*a^344 - 1/48*a^340 - 1/96*a^332 + 1/960*a^316)*s)*t^7 + ((-1/4*a^364 + 5/4*a^360 - 19/8*a^356 + 27/16*a^352 + 3/16*a^348 - 11/16*a^344 + 1/8*a^340 + 1/48*a^336 + 7/96*a^332 - 1/24*a^328 - 1/80*a^324 + 1/40*a^320 - 3/320*a^316)*s)*t^6 + ((1/2*a^364 - 3*a^360 + 59/8*a^356 - 37/4*a^352 + 17/3*a^348 - 2/3*a^344 - 7/6*a^340 + 3/4*a^336 - 1/3*a^332 + 1/6*a^328 - 3/80*a^324 - 1/60*a^320 + 1/40*a^316 - 1/60*a^312 + 1/240*a^308)*s^2 + (-7/16*a^368 + 211/96*a^364 - 27/8*a^360 - 43/48*a^356 + 395/48*a^352 - 279/32*a^348 + 5/2*a^344 + 39/32*a^340 - 9/16*a^336 - 61/192*a^332 - 1/48*a^328 + 73/160*a^324 - 67/240*a^320 - 1/160*a^316 + 3/80*a^312 - 1/240*a^308)*s)*t^5 + ((a^368 - 27/4*a^364 + 75/4*a^360 - 317/12*a^356 + 67/4*a^352 + 5/2*a^348 - 67/6*a^344 + 51/8*a^340 - 11/12*a^336 + 11/24*a^332 - a^328 + 3/16*a^324 + 7/12*a^320 - 1/2*a^316 + 1/6*a^312 - 1/48*a^308)*s^2 + (1/96*a^380 - 1/8*a^376 + 1/8*a^372 + 11/12*a^368 - 193/192*a^364 - 20/3*a^360 + 469/24*a^356 - 935/48*a^352 + 31/12*a^348 + 445/48*a^344 - 247/48*a^340 - 29/16*a^336 + 119/96*a^332 + 39/16*a^328 - 125/48*a^324 + 5/24*a^320 + 73/96*a^316 - 3/16*a^312 - 1/6*a^308 + 1/16*a^304)*s)*t^4 + ((1/4*a^376 - 17/24*a^372 - 8/3*a^368 + 199/12*a^364 - 69/2*a^360 + 823/24*a^356 - 133/12*a^352 - 241/24*a^348 + 25/2*a^344 - 415/48*a^340 + 127/12*a^336 - 89/8*a^332 + 31/12*a^328 + 22/3*a^324 - 103/12*a^320 + 11/3*a^316 - 1/6*a^312 - 17/48*a^308 + 1/12*a^304)*s^2 + (1/960*a^396 - 1/48*a^388 + 1/24*a^384 - 3/32*a^380 + 5/24*a^376 + 11/32*a^372 - 9/16*a^368 - 53/8*a^364 + 557/24*a^360 - 737/24*a^356 + 673/48*a^352 + 749/96*a^348 - 269/24*a^344 + 147/32*a^340 - 11/2*a^336 + 1019/96*a^332 - 109/16*a^328 - 133/32*a^324 + 209/24*a^320 - 59/15*a^316 - 25/24*a^312 + 37/24*a^308 - 17/48*a^304 + 1/32*a^300 - 3/16*a^296 + 1/8*a^292)*s)*t^3 + ((1/24*a^388 - 1/6*a^384 + 1/3*a^380 - a^376 + 89/48*a^372 + 29/12*a^368 - 129/8*a^364 + 337/12*a^360 - 363/16*a^356 + 41/6*a^352 + 25/24*a^348 - 7/4*a^344 + 31/4*a^340 - 187/12*a^336 + 43/4*a^332 + 14/3*a^328 - 193/16*a^324 + 23/4*a^320 + 1/3*a^316 + 25/12*a^312 - 93/16*a^308 + 19/4*a^304 - 7/4*a^300 + 1/4*a^296)*s^2 + (-1/480*a^404 + 1/240*a^400 - 1/240*a^396 + 1/12*a^384 - 17/96*a^380 + 11/24*a^376 - 143/96*a^372 + 1/48*a^368 + 1909/192*a^364 - 45/2*a^360 + 1025/48*a^356 - 59/8*a^352 - 13/12*a^348 + 1/2*a^344 - 63/16*a^340 + 305/24*a^336 - 25/2*a^332 - 15/16*a^328 + 5401/480*a^324 - 557/80*a^320 - 157/160*a^316 + 17/48*a^312 + 197/48*a^308 - 191/48*a^304 + 61/96*a^300 + a^296 - 23/48*a^292 - 1/16*a^288 - 3/32*a^284 + 1/4*a^280 - 1/8*a^276)*s)*t^2 + ((1/240*a^404 - 1/60*a^400 + 1/40*a^396 - 1/60*a^392 - 3/80*a^388 + 1/6*a^384 - 1/3*a^380 + 3/4*a^376 - 7/6*a^372 - 2/3*a^368 + 17/3*a^364 - 37/4*a^360 + 89/12*a^356 - 13/4*a^352 + 19/24*a^348 + 4/3*a^344 - 235/48*a^340 + 35/6*a^336 - 1/6*a^332 - 37/6*a^328 + 547/120*a^324 + 19/15*a^320 - 143/120*a^316 - 259/60*a^312 + 1649/240*a^308 - 16/3*a^304 + 16/3*a^300 - 85/12*a^296 + 49/8*a^292 - 11/4*a^288 + 1/2*a^284)*s^2 + (-1/480*a^404 + 1/80*a^400 - 7/320*a^396 + 1/60*a^392 + 1/60*a^388 - 1/8*a^384 + 25/96*a^380 - 13/24*a^376 + 25/24*a^372 - 1/48*a^368 - 397/96*a^364 + 8*a^360 - 337/48*a^356 + 19/6*a^352 - 79/96*a^348 - 11/16*a^344 + 15/4*a^340 - 265/48*a^336 + 253/192*a^332 + 41/8*a^328 - 393/80*a^324 - 169/240*a^320 + 353/192*a^316 + 247/80*a^312 - 371/60*a^308 + 77/16*a^304 - 415/96*a^300 + 299/48*a^296 - 283/48*a^292 + 17/8*a^288 + 9/8*a^284 - 3/2*a^280 + 1/2*a^276)*s)*t + (-1/240*a^324 + 1/60*a^320 - 1/40*a^316 + 1/60*a^312 + 3/80*a^308 - 1/6*a^304 + 1/3*a^300 - 3/4*a^296 + 7/6*a^292 + 2/3*a^288 - 17/3*a^284 + 37/4*a^280 - 59/8*a^276 + 3*a^272 - 1/2*a^268)*s^2 + (1/480*a^324 - 1/80*a^320 + 7/320*a^316 - 1/60*a^312 - 1/60*a^308 + 1/8*a^304 - 25/96*a^300 + 13/24*a^296 - 25/24*a^292 + 1/48*a^288 + 397/96*a^284 - 8*a^280 + 7*a^276 - 3*a^272 + 1/2*a^268)*s)*w^7 + (((-1/48*a^312 + 1/48*a^308 - 1/288*a^300)*s)*t^6 + ((1/8*a^320 - 19/48*a^316 + 23/48*a^312 - 17/96*a^308 - 1/16*a^304 + 1/24*a^300 + 1/48*a^288 - 1/96*a^284)*s)*t^5 + ((-1/4*a^320 + 5/4*a^316 - 5/2*a^312 + 59/24*a^308 - 13/12*a^304 + 1/6*a^296 - 1/24*a^292)*s^2 + (-1/12*a^336 + 1/12*a^332 + 1/2*a^328 - 257/288*a^324 - 1/4*a^320 + 53/48*a^316 + 13/24*a^312 - 55/24*a^308 + 19/12*a^304 - 13/72*a^300 - 7/24*a^296 + 35/96*a^292 - 1/3*a^288 + 5/48*a^284 + 1/96*a^268 - 1/48*a^264)*s)*t^4 + ((1/6*a^348 - 1/6*a^344 - 47/36*a^336 + 1/3*a^332 + 7*a^328 - 353/36*a^324 + 59/12*a^320 - 59/8*a^316 + 421/36*a^312 - 7/3*a^308 - 17/2*a^304 + 167/24*a^300 - a^296 - 13/8*a^292 + 13/6*a^288 - 7/4*a^284 + 3/4*a^280 - 1/8*a^276)*s^2 + (-1/6*a^348 + 1/6*a^344 - 1/12*a^340 + 14/9*a^336 + 1/16*a^332 - 73/8*a^328 + 3317/288*a^324 - 29/12*a^320 + 37/16*a^316 - 671/72*a^312 + 55/16*a^308 + 53/8*a^304 - 71/12*a^300 + 3/2*a^296 - 1/32*a^292 - 7/8*a^288 + 17/12*a^284 - 3/4*a^280 + 1/16*a^276 + 1/4*a^272 - 11/32*a^268 + 3/16*a^264 + 1/288*a^252 - 1/48*a^244 + 1/48*a^240)*s)*t^3 + ((1/6*a^352 - 1/3*a^348 + 1/12*a^344 - 7/6*a^340 + 19/12*a^336 + 175/24*a^332 - 17*a^328 + 113/12*a^324 - 7/6*a^320 + 69/8*a^316 - 19/4*a^312 - 103/8*a^308 + 34/3*a^304 + 53/24*a^300 - 41/6*a^296 + 223/24*a^292 - 35/4*a^288 + 5/8*a^284 + 9/2*a^280 - 21/8*a^276 - 1/4*a^272 + 7/4*a^268 - 2*a^264 + 9/8*a^260 - 1/4*a^256)*s^2 + (-1/6*a^352 + 23/72*a^348 - 1/12*a^344 + 17/12*a^340 - 23/12*a^336 - 67/8*a^332 + 119/6*a^328 - 2935/288*a^324 - 17/6*a^320 - 63/16*a^316 + 65/16*a^312 + 505/48*a^308 - 103/12*a^304 - 1087/288*a^300 + 143/24*a^296 - 655/96*a^292 + 7*a^288 - 23/48*a^284 - 15/4*a^280 + 19/8*a^276 - 3/4*a^272 - 49/96*a^268 + 73/48*a^264 - 55/48*a^260 + 7/24*a^256 + 3/32*a^252 - 3/8*a^248 + 7/16*a^244 - 3/16*a^240)*s)*t^2 + ((1/36*a^360 - 1/6*a^352 - 1/18*a^348 - 1/12*a^344 + 13/6*a^340 - 5/6*a^336 - 61/8*a^332 + 28/3*a^328 + 13/18*a^324 - 17/6*a^320 - 5/2*a^316 - 34/9*a^312 + 39/4*a^308 - 1/6*a^304 - 9*a^300 + 113/12*a^296 - 191/24*a^292 + 15/4*a^288 - 7/8*a^284 + 23/12*a^280 - 7/12*a^276 + 2*a^272 - 25/4*a^268 + 2*a^264 + 59/12*a^260 - 53/12*a^256 + a^252 + 13/12*a^248 - 59/24*a^244 + 5/2*a^240 - 5/4*a^236 + 1/4*a^232)*s^2 + (-1/36*a^360 + 1/6*a^352 + 5/72*a^348 + 1/12*a^344 - 7/3*a^340 + a^336 + 395/48*a^332 - 253/24*a^328 - 221/288*a^324 + 113/24*a^320 + 11/12*a^316 + 517/144*a^312 - 817/96*a^308 - 55/48*a^304 + 29/3*a^300 - 107/12*a^296 + 655/96*a^292 - 143/48*a^288 + 31/32*a^284 - 8/3*a^280 + 131/144*a^276 - 5/4*a^272 + 497/96*a^268 - 73/48*a^264 - 13/3*a^260 + 13/4*a^256 - 265/288*a^252 + 5/12*a^248 + 47/48*a^244 - 97/48*a^240 + 5/4*a^236 - 1/4*a^232)*s)*t + (-1/36*a^288 + 1/6*a^280 + 1/18*a^276 + 1/6*a^272 - 7/3*a^268 + 8/9*a^264 + 167/24*a^260 - 103/12*a^256 + 3*a^252 - 21/4*a^248 + 69/8*a^244 - 2/3*a^240 - 27/4*a^236 + 19/4*a^232 - a^228)*s^2 + (1/36*a^288 - 1/6*a^280 - 5/72*a^276 - 1/6*a^272 + 5/2*a^268 - 19/18*a^264 - 361/48*a^260 + 233/24*a^256 - 305/96*a^252 + 33/8*a^248 - 121/16*a^244 + 17/48*a^240 + 27/4*a^236 - 19/4*a^232 + a^228)*s)*w^6 + (1/384*a^252*s*t^5 + ((-1/32*a^260 + 1/16*a^256 - 1/24*a^252 - 1/384*a^220)*s)*t^4 + ((1/16*a^260 - 1/4*a^256 + 3/8*a^252 - 1/4*a^248 + 1/16*a^244)*s^2 + (1/24*a^276 - 7/32*a^268 + 1/8*a^264 + 11/32*a^260 - 5/16*a^256 - 29/192*a^252 + 1/4*a^248 - 1/16*a^244 + 1/32*a^228 - 1/16*a^224 + 1/24*a^220)*s)*t^3 + ((-1/12*a^288 + 2/3*a^276 + 1/2*a^272 - 25/8*a^268 + 29/12*a^264 - 2*a^260 + 19/4*a^256 - 29/8*a^252 - 3/4*a^248 + 7/4*a^244 - 1/2*a^240 - 1/16*a^228 + 1/4*a^224 - 3/8*a^220 + 1/4*a^216 - 1/16*a^212)*s^2 + (-1/48*a^300 + 1/4*a^288 + 1/8*a^284 - 3/4*a^280 - 1/3*a^276 - 1/2*a^272 + 4*a^268 - 35/12*a^264 + 25/32*a^260 - 53/16*a^256 + 19/6*a^252 + 3/4*a^248 - 43/24*a^244 + 1/2*a^240 + 7/32*a^236 - 1/8*a^232 - 11/32*a^228 + 5/16*a^224 + 29/192*a^220 - 1/4*a^216 + 1/16*a^212)*s)*t^2 + ((1/8*a^268 - a^264 + 7/2*a^260 - 7*a^256 + 35/4*a^252 - 7*a^248 + 7/2*a^244 - a^240 + 1/8*a^236)*s^3 + (1/24*a^332 - a^308 + 3*a^296 + 6*a^292 - 191/12*a^288 + 17/2*a^284 - 2/3*a^276 - 9/2*a^272 - 1/4*a^268 + 223/12*a^264 - 899/48*a^260 + 163/12*a^256 - 91/4*a^252 + 17*a^248 + 25/48*a^244 - 6*a^240 + 37/8*a^236 - 29/12*a^232 + 2*a^228 - 19/4*a^224 + 29/8*a^220 + 3/4*a^216 - 7/4*a^212 + 1/2*a^208)*s^2 + (-1/24*a^332 + a^308 + 1/48*a^300 - 3*a^296 - 6*a^292 + 63/4*a^288 - 69/8*a^284 + 3/4*a^280 + 7/24*a^276 + 9/2*a^272 - 49/96*a^268 - 413/24*a^264 + 1543/96*a^260 - 123/16*a^256 + 1811/128*a^252 - 37/4*a^248 - 209/48*a^244 + 7*a^240 - 45/8*a^236 + 35/12*a^232 - 25/32*a^228 + 53/16*a^224 - 19/6*a^220 - 3/4*a^216 + 7/4*a^212 - 1/2*a^208)*s)*t + (-1/8*a^236 + a^232 - 7/2*a^228 + 7*a^224 - 35/4*a^220 + 7*a^216 - 7/2*a^212 + a^208 - 1/8*a^204)*s^3 + (-1/24*a^300 + a^276 - 3*a^264 - 6*a^260 + 191/12*a^256 - 17/2*a^252 + 2/3*a^244 + 9/2*a^240 + 1/4*a^236 - 223/12*a^232 + 899/48*a^228 - 27/2*a^224 + 91/4*a^220 - 17*a^216 - 19/16*a^212 + 11/2*a^208 - 3/2*a^204)*s^2 + (1/24*a^300 - a^276 - 1/48*a^268 + 3*a^264 + 6*a^260 - 63/4*a^256 + 69/8*a^252 - 3/4*a^248 - 7/24*a^244 - 9/2*a^240 + 17/32*a^236 + 413/24*a^232 - 1543/96*a^228 + 119/16*a^224 - 1827/128*a^220 + 10*a^216 + 75/16*a^212 - 13/2*a^208 + 13/8*a^204)*s)*w^5]" ] }, "execution_count": 293, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNF (we take a parameter a = sqrt{sqrt{2}})\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "QCNFzero = [qcnf[i] * a^(i*(i-1)) for i in srange(N)]\n", "QCNFzero" ] }, { "cell_type": "code", "execution_count": 294, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 8*s*t - 8*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " -16*s*t + 16*s,\n", " 192*s*t^2 - 192*s*t,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " -2048*s*t^2 + 2560*s*t - 512*s,\n", " 51200/3*s*t^3 - 10240*s*t^2 + 4096*s*t - 32768/3*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 2048*s*t^2 + 2048*s^2 - 8192*s*t + 4096*s,\n", " -786432*s*t^3 + 786432*s*t^2,\n", " 5931008*s*t^4 - 851968/3*s*t^3 + 5472256*s*t^2 + 16973824/3*s*t - 16777216*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 4194304*s*t^3 - 17039360*s*t^2 - 262144*s^2 + 1048576*(4*s^2 + 9*s)*t - 524288*s,\n", " -3355443200/3*s*t^4 + 1744830464/3*s*t^3 - 134217728*s*t^2 + 2046820352/3*s*t - 33554432/3*s,\n", " 122813415424/15*s*t^5 + 13748928512/3*s*t^4 + 16965959680*s*t^3 + 123505475584/3*s*t^2 + 84859158528*s*t - 2336462209024/15*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -8388608/3*s*t^3 + 25165824*s*t^2 + 13497270272/3*s^2 - 8388608*(s^2 + 28*s)*t - 4278190080*s,\n", " 25769803776*s*t^4 - 96636764160*s*t^3 + 25769803776*(s^2 + s)*t^2 + 6442450944*(s^2 + 2*s)*t,\n", " -6219112644608*s*t^5 - 1065151889408/3*s*t^4 - 14809047236608/3*s*t^3 - 15427522527232/3*s*t^2 + 50165218017280/3*s*t - 68719476736*s,\n", " 225575440482304/5*s*t^6 + 803299007660032/15*s*t^5 + 496140126519296/3*s*t^4 + 4602751631753216/9*s*t^3 + 5580082714247168/3*s*t^2 + 274060415753781248/45*s*t - 8725724278030336*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " -274877906944/3*s*t^4 + 2478196129792/3*s*t^3 - 4294967296*(64*s^2 + 1795*s)*t^2 - 6910602379264/3*s^2 + 4294967296/3*(102979*s^2 - 97836*s)*t + 2190433320960*s,\n", " 1759218604441600/3*s*t^5 - 5717460464435200/3*s*t^4 + 17592186044416/3*(100*s^2 - 31*s)*t^3 + 1374389534720/3*(960*s^2 - 1613*s)*t^2 - 274877906944/3*s^2 + 1099511627776/3*(1104*s^2 + 2209*s)*t - 549755813888/3*s,\n", " -2060467198266179584/15*s*t^6 - 461830068038008832/5*s*t^5 - 818318126042054656/3*s*t^4 - 1980669042368708608/3*s*t^3 - 4003559331244015616/3*s*t^2 + 37496970497486749696/15*s*t - 38280596832649216/15*s,\n", " 62605052166885343232/63*s*t^7 + 27145815915004690432/15*s*t^6 + 16944390901847818240/3*s*t^5 + 176930388309371256832/9*s*t^4 + 259844603315790807040/3*s*t^3 + 21198940523145214296064/45*s*t^2 + 37260710258058056433664/15*s*t - 967005994459965260038144/315*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 137438953472/3*s*t^4 - 2199023255552/3*s*t^3 + 137438953472*s^3 + 274877906944*(s^2 + 56*s)*t^2 + 1768905027004399616/3*s^2 - 2199023255552/3*(403*s^2 + 440*s)*t - 1767096605254615040/3*s,\n", " -9007199254740992*s*t^5 + 78250043525562368*s*t^4 - 1688849860263936*(16*s^2 + 433*s)*t^3 + 562949953421312*(25729*s^2 - 24900*s)*t^2 + 2814749767106560*(1609*s^2 - 1530*s)*t,\n", " 52169698083459825664*s*t^6 - 417357584667678605312/3*s*t^5 + 288230376151711744/3*(543*s^2 - 679*s)*t^4 + 36028797018963968/3*(5792*s^2 - 12049*s)*t^3 + 2251799813685248/3*(144768*s^2 - 514243*s)*t^2 - 2251799813685248*s^2 + 9007199254740992/3*(50236*s^2 + 100475*s)*t - 4503599627370496*s,\n", " -60552446228268134170624/5*s*t^7 - 236528764199122298208256/15*s*t^6 - 220532554783454599839744/5*s*t^5 - 1212718643134409549545472/9*s*t^4 - 4366799870356558647918592/9*s*t^3 - 70821290079573784181866496/45*s*t^2 + 101983967156714321494409216/45*s*t - 571849066284996100096*s,\n", " 9191358120829490972065792/105*s*t^8 + 67765037449565623155687424/315*s*t^7 + 10656527504218266988969984/15*s*t^6 + 119967100385894431385976832/45*s*t^5 + 38556242257361751920082944/3*s*t^4 + 3774000931040806702271168512/45*s*t^3 + 10744940266627128213706375168/15*s*t^2 + 219624751296062137230864416768/35*s*t - 106375800719530450340745838592/15*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 72057594037927936/3*s*t^5 - 1153202979583557632/3*s*t^4 + 4503599627370496/3*(96*s^2 + 5377*s)*t^3 - 281474976710656*s^3 - 562949953421312/3*(825347*s^2 + 901288*s)*t^2 - 3622717495305010413568/3*s^2 + 4503599627370496/3*(48*s^3 + 205928051*s^2 - 205716680*s)*t + 3619013847561451601920/3*s,\n", " -29514790517935282585600/9*s*t^6 + 80501591137668483252224/3*s*t^5 - 147573952589676412928/3*(200*s^2 + 5157*s)*t^4 + 72057594037927936/9*(658041856*s^2 - 652188671*s)*t^3 + 72057594037927936/3*(179298304*s^2 - 178944003*s)*t^2 - 115940668807026049024/9*s^2 + 72057594037927936/3*(156523521*s^2 - 148838372*s)*t + 12249790986447749120*s,\n", " 276551225969812163004465152/15*s*t^7 - 188922993513683026419122176/5*s*t^6 + 9444732965739290427392/15*(29281*s^2 - 51173*s)*t^5 + 4722366482869645213696/3*(22846*s^2 - 37901*s)*t^4 + 1180591620717411303424/3*(189960*s^2 - 430679*s)*t^3 + 295147905179352825856/15*(8509360*s^2 - 54283137*s)*t^2 - 5017514388048998039552/15*s^2 + 1180591620717411303424/15*(4456365*s^2 + 8912747*s)*t - 10035028776097996079104/15*s,\n", " -268886651621146483363174940672/63*s*t^8 - 2604006978890043272817325113344/315*s*t^7 - 366843444528624206717403529216/15*s*t^6 - 755649196497947030995320438784/9*s*t^5 - 3302452068797962966543017967616/9*s*t^4 - 89362039331954348521991125860352/45*s*t^3 - 468965923107996655934267444953088/45*s*t^2 + 4062234928756522478271477953069056/315*s*t - 253494819411713133127439220736/315*s,\n", " 87437743799455020655277680623616/2835*s*t^9 + 30066662019989192360234240180224/315*s*t^8 + 5069267410930467537214800658432/15*s*t^7 + 60426903408142079853725803872256/45*s*t^6 + 306082061004930337002651436187648/45*s*t^5 + 6498215039911045734825617401053184/135*s*t^4 + 22220055489907002416758215487258624/45*s*t^3 + 2187804720187187133173061069955923968/315*s*t^2 + 98596953246954360933505063400863956992/945*s*t - 317041834428250708890138228122605060096/2835*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0]]" ] }, "execution_count": 294, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNF (Table 24)\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "QCNFzeroMatrix = [[QCNFzero[i][j].subs(a=sqrt(sqrt(2))) for j in srange(N)] for i in srange(N)]\n", "QCNFzeroMatrix" ] }, { "cell_type": "code", "execution_count": 295, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -8*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 16*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -512*s, -32768/3*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 2048*s^2 + 4096*s,\n", " 0,\n", " -16777216*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -262144*s^2 - 524288*s,\n", " -33554432/3*s,\n", " -2336462209024/15*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 13497270272/3*s^2 - 4278190080*s,\n", " 0,\n", " -68719476736*s,\n", " -8725724278030336*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " -6910602379264/3*s^2 + 2190433320960*s,\n", " -274877906944/3*s^2 - 549755813888/3*s,\n", " -38280596832649216/15*s,\n", " -967005994459965260038144/315*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 137438953472*s^3 + 1768905027004399616/3*s^2 - 1767096605254615040/3*s,\n", " 0,\n", " -2251799813685248*s^2 - 4503599627370496*s,\n", " -571849066284996100096*s,\n", " -106375800719530450340745838592/15*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " -281474976710656*s^3 - 3622717495305010413568/3*s^2 + 3619013847561451601920/3*s,\n", " -115940668807026049024/9*s^2 + 12249790986447749120*s,\n", " -5017514388048998039552/15*s^2 - 10035028776097996079104/15*s,\n", " -253494819411713133127439220736/315*s,\n", " -317041834428250708890138228122605060096/2835*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0]]" ] }, "execution_count": 295, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNF with no ordinary components, t=0 (Table 25)\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "QCNFzeroMatrixZeroT = [[QCNFzeroMatrix[i][j].subs(t=0) for j in srange(N)] for i in srange(N)]\n", "QCNFzeroMatrixZeroT" ] }, { "cell_type": "code", "execution_count": 296, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -512, -32768/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 4096, 0, -16777216, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -524288,\n", " -33554432/3,\n", " -2336462209024/15,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -4278190080,\n", " 0,\n", " -68719476736,\n", " -8725724278030336,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 2190433320960,\n", " -549755813888/3,\n", " -38280596832649216/15,\n", " -967005994459965260038144/315,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " -1767096605254615040/3,\n", " 0,\n", " -4503599627370496,\n", " -571849066284996100096,\n", " -106375800719530450340745838592/15,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 3619013847561451601920/3,\n", " 12249790986447749120,\n", " -10035028776097996079104/15,\n", " -253494819411713133127439220736/315,\n", " -317041834428250708890138228122605060096/2835,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0]]" ] }, "execution_count": 296, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNF with no ordinary components (t=0) and one contradictory component (Table 23)\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "QCNFzeroMatrixZeroTDS = [[QCNFzeroMatrix[i][j].diff(s).subs(s=0,t=0) for j in srange(N)] for i in srange(N)]\n", "QCNFzeroMatrixZeroTDS" ] }, { "cell_type": "code", "execution_count": 298, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 2048, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, -262144, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 13497270272/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " -6910602379264/3,\n", " -274877906944/3,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 1768905027004399616/3,\n", " 0,\n", " -2251799813685248,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " -3622717495305010413568/3,\n", " -115940668807026049024/9,\n", " -5017514388048998039552/15,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0]]" ] }, "execution_count": 298, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNF with no ordinary components (t=0) and two contradictory components (Table 26)\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "QCNFzeroMatrixZeroTDDS = [[QCNFzeroMatrix[i][j].diff(s,2).subs(s=0,t=0)/2 for j in srange(N)] for i in srange(N)]\n", "QCNFzeroMatrixZeroTDDS" ] }, { "cell_type": "code", "execution_count": 299, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 8*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -16*s, -192*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 2560*s, 4096*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -8192*s, 0, 16973824/3*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 4194304*s^2 + 9437184*s,\n", " 2046820352/3*s,\n", " 84859158528*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -8388608*s^2 - 234881024*s,\n", " 6442450944*s^2 + 12884901888*s,\n", " 50165218017280/3*s,\n", " 274060415753781248/45*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 442291437174784/3*s^2 - 140067473457152*s,\n", " 404620279021568*s^2 + 2428821185757184/3*s,\n", " 37496970497486749696/15*s,\n", " 37260710258058056433664/15*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " -886206371987456/3*s^2 - 967570232442880/3*s,\n", " 4528932375274455040*s^2 - 4306567143673036800*s,\n", " 452485661761168474112/3*s^2 + 904998345120101171200/3*s,\n", " 101983967156714321494409216/45*s,\n", " 219624751296062137230864416768/35*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 72057594037927936*s^3 + 927417493748732496183296/3*s^2 - 926465563391895567073280/3*s,\n", " 3759569444535029362327552*s^2 - 10724934986842100247560192/3*s,\n", " 350743145190556441545539584*s^2 + 10522314425774245442358345728/15*s,\n", " 4062234928756522478271477953069056/315*s,\n", " 98596953246954360933505063400863956992/945*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0]]" ] }, "execution_count": 299, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNF with one pair of ordinary components (Table 27)\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "QCNFzeroMatrixDT = [[QCNFzeroMatrix[i][j].diff(t).subs(t=0) for j in srange(N)] for i in srange(N)]\n", "QCNFzeroMatrixDT" ] }, { "cell_type": "code", "execution_count": 302, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 192*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -2048*s, -10240*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 2048*s,\n", " 786432*s,\n", " 5472256*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -17039360*s,\n", " -134217728*s,\n", " 123505475584/3*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 25165824*s,\n", " 25769803776*s^2 + 25769803776*s,\n", " -15427522527232/3*s,\n", " 5580082714247168/3*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " -274877906944*s^2 - 7709466296320*s,\n", " 439804651110400*s^2 - 2216890319503360/3*s,\n", " -4003559331244015616/3*s,\n", " 21198940523145214296064/45*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 274877906944*s^2 + 15393162788864*s,\n", " 14484139351576936448*s^2 - 14017453840190668800*s,\n", " 108662851809195327488*s^2 - 1157972291588942987264/3*s,\n", " -70821290079573784181866496/45*s,\n", " 10744940266627128213706375168/15*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " -464629055206419595264/3*s^2 - 507380037619187449856/3*s,\n", " 12919804401320990599020544/3*s^2 - 4298091441231919564455936*s,\n", " 502303955683395552445202432/3*s^2 - 5340518057371273005759463424/5*s,\n", " -468965923107996655934267444953088/45*s,\n", " 2187804720187187133173061069955923968/315*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0]]" ] }, "execution_count": 302, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNF with two pairs of ordinary components (Table 28)\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "QCNFzeroMatrixDT = [[QCNFzeroMatrix[i][j].diff(t,2).subs(t=0)/2 for j in srange(N)] for i in srange(N)]\n", "QCNFzeroMatrixDT" ] }, { "cell_type": "code", "execution_count": 304, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 51200/3*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -786432*s,\n", " -851968/3*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 4194304*s,\n", " 1744830464/3*s,\n", " 16965959680*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -8388608/3*s,\n", " -96636764160*s,\n", " -14809047236608/3*s,\n", " 4602751631753216/9*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 2478196129792/3*s,\n", " 1759218604441600/3*s^2 - 545357767376896/3*s,\n", " -1980669042368708608/3*s,\n", " 259844603315790807040/3*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " -2199023255552/3*s,\n", " -27021597764222976*s^2 - 731271989494284288*s,\n", " 208678792333839302656/3*s^2 - 434110975281496850432/3*s,\n", " -4366799870356558647918592/9*s,\n", " 3774000931040806702271168512/45*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 144115188075855872*s^2 + 24215855196371156992/3*s,\n", " 47416912919612633399689216/9*s^2 - 46995146491053744173613056/9*s,\n", " 74755061423826483732807680*s^2 - 508456018618953982747344896/3*s,\n", " -89362039331954348521991125860352/45*s,\n", " 22220055489907002416758215487258624/45*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0]]" ] }, "execution_count": 304, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNF with three pairs of ordinary components (Table 29)\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "QCNFzeroMatrixDT = [[QCNFzeroMatrix[i][j].diff(t,3).subs(t=0)/6 for j in srange(N)] for i in srange(N)]\n", "QCNFzeroMatrixDT" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", " Verification of the last asymptotics\n", "" ] }, { "cell_type": "code", "execution_count": 220, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 0,\n", " -2*a^11*w,\n", " 0,\n", " (-12*a^30 + 24*a^26)*w^2 + 12*a^14*w,\n", " 0,\n", " (-120*a^57 + 720*a^49 - 720*a^45)*w^3 - 360*a^25*w^2,\n", " 0,\n", " (-1680*a^92 + 13440*a^80 + 10080*a^76 - 60480*a^72 + 40320*a^68)*w^4 + (-10080*a^44 + 20160*a^40)*w^3 + (-10080*a^36 + 60480*a^32 - 105840*a^28 + 80640*a^24 - 20160*a^20)*w^2,\n", " 0,\n", " (-30240*a^135 + 302400*a^119 + 604800*a^111 - 1814400*a^107 - 2721600*a^103 + 7257600*a^99 - 3628800*a^95)*w^5 + (-302400*a^71 + 1814400*a^63 - 1814400*a^59)*w^4 + (907200*a^47 - 5443200*a^43 + 9525600*a^39 - 7257600*a^35 + 1814400*a^31)*w^3,\n", " 0,\n", " (-665280*a^186 + 7983360*a^166 + 19958400*a^154 - 46569600*a^150 - 239500800*a^142 + 259459200*a^138 + 718502400*a^134 - 1197504000*a^130 + 479001600*a^126)*w^6 + (-9979200*a^106 + 79833600*a^94 + 59875200*a^90 - 359251200*a^86 + 239500800*a^82)*w^5 + (59875200*a^66 - 479001600*a^62 + 1347192000*a^58 - 1736380800*a^54 + 1077753600*a^50 - 239500800*a^46)*w^4 + (-79833600*a^78 + 678585600*a^66 + 479001600*a^62 - 3113510400*a^58 + 1836172800*a^54 + 59875200*a^50 + 1796256000*a^46 - 1526817600*a^42 - 1437004800*a^38 + 1796256000*a^34 - 479001600*a^30)*w^3,\n", " 0,\n", " (-17297280*a^245 + 242161920*a^221 + 726485760*a^205 - 2179457280*a^201 + 1210809600*a^197 - 10897286400*a^189 + 7264857600*a^185 - 10897286400*a^181 + 87178291200*a^177 - 29059430400*a^173 - 217945728000*a^169 + 261534873600*a^165 - 87178291200*a^161)*w^7 + (-363242880*a^149 + 3632428800*a^133 + 7264857600*a^125 - 21794572800*a^121 - 32691859200*a^117 + 87178291200*a^113 - 43589145600*a^109)*w^6 + (3632428800*a^93 - 21794572800*a^89 + 16345929600*a^85 + 123502579200*a^81 - 352345593600*a^77 + 403199596800*a^73 - 217945728000*a^69 + 43589145600*a^65)*w^5 + (14529715200*a^89 - 123502579200*a^77 - 87178291200*a^73 + 566658892800*a^69 - 334183449600*a^65 - 10897286400*a^61 - 326918592000*a^57 + 277880803200*a^53 + 261534873600*a^49 - 326918592000*a^45 + 87178291200*a^41)*w^4]" ] }, "execution_count": 220, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# CGF of contradictory strongly connected implication digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "q1 = exp(scd.subs(z = a^(14)*z^4*w)/2 - cscc.subs(z = a^(10)*z^4*w)) * exp_had_prod(PhiTwoTwoFour,1/g.subs(z = a^(10)*z^2*w),N)\n", "Q1 = [q1[i] * i.factorial() for i in srange(N)]\n", "Q1" ] }, { "cell_type": "code", "execution_count": 221, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1,\n", " 0,\n", " -a^12*w,\n", " 0,\n", " (-1/2*a^36 + a^32)*w^2 + 1/2*a^20*w,\n", " 0,\n", " (-1/6*a^72 + a^64 - a^60)*w^3 - 1/2*a^40*w^2,\n", " 0,\n", " (-1/24*a^120 + 1/3*a^108 + 1/4*a^104 - 3/2*a^100 + a^96)*w^4 + (-1/4*a^72 + 1/2*a^68)*w^3 + (-1/4*a^64 + 3/2*a^60 - 21/8*a^56 + 2*a^52 - 1/2*a^48)*w^2,\n", " 0,\n", " (-1/120*a^180 + 1/12*a^164 + 1/6*a^156 - 1/2*a^152 - 3/4*a^148 + 2*a^144 - a^140)*w^5 + (-1/12*a^116 + 1/2*a^108 - 1/2*a^104)*w^4 + (1/4*a^92 - 3/2*a^88 + 21/8*a^84 - 2*a^80 + 1/2*a^76)*w^3,\n", " 0,\n", " (-1/720*a^252 + 1/60*a^232 + 1/24*a^220 - 7/72*a^216 - 1/2*a^208 + 13/24*a^204 + 3/2*a^200 - 5/2*a^196 + a^192)*w^6 + (-1/48*a^172 + 1/6*a^160 + 1/8*a^156 - 3/4*a^152 + 1/2*a^148)*w^5 + (1/8*a^132 - a^128 + 45/16*a^124 - 29/8*a^120 + 9/4*a^116 - 1/2*a^112)*w^4 + (-1/6*a^144 + 17/12*a^132 + a^128 - 13/2*a^124 + 23/6*a^120 + 1/8*a^116 + 15/4*a^112 - 51/16*a^108 - 3*a^104 + 15/4*a^100 - a^96)*w^3,\n", " 0,\n", " (-1/5040*a^336 + 1/360*a^312 + 1/120*a^296 - 1/40*a^292 + 1/72*a^288 - 1/8*a^280 + 1/12*a^276 - 1/8*a^272 + a^268 - 1/3*a^264 - 5/2*a^260 + 3*a^256 - a^252)*w^7 + (-1/240*a^240 + 1/24*a^224 + 1/12*a^216 - 1/4*a^212 - 3/8*a^208 + a^204 - 1/2*a^200)*w^6 + (1/24*a^184 - 1/4*a^180 + 3/16*a^176 + 17/12*a^172 - 97/24*a^168 + 37/8*a^164 - 5/2*a^160 + 1/2*a^156)*w^5 + (1/6*a^180 - 17/12*a^168 - a^164 + 13/2*a^160 - 23/6*a^156 - 1/8*a^152 - 15/4*a^148 + 51/16*a^144 + 3*a^140 - 15/4*a^136 + a^132)*w^4]" ] }, "execution_count": 221, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of contradictory strongly connected implication digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "Q1zero = [q1[i] * a^(i*(i-1)/2) for i in srange(N)]\n", "Q1zero" ] }, { "cell_type": "code", "execution_count": 222, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -512, -32768/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 4096, 0, -16777216, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, -524288, -33554432/3, -2336462209024/15, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -4278190080,\n", " 0,\n", " -68719476736,\n", " -8725724278030336,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 0,\n", " 2190433320960,\n", " -549755813888/3,\n", " -38280596832649216/15,\n", " -967005994459965260038144/315,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0]]" ] }, "execution_count": 222, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of contradictory strongly connected implication digraphs (Table 23)\n", "Q1zeroMatrix = [[Q1zero[i][j].subs(a=sqrt(sqrt(2))) for j in srange(N)] for i in srange(N)]\n", "Q1zeroMatrix" ] }, { "cell_type": "code", "execution_count": 223, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " 1,\n", " 0,\n", " -6*a^6*w,\n", " 0,\n", " (-60*a^16 + 120*a^12)*w^2 + 60*w,\n", " 0,\n", " (-840*a^30 + 5040*a^22 - 5040*a^18)*w^3 - 2520/a^2*w^2,\n", " 0,\n", " (-15120*a^48 + 120960*a^36 + 90720*a^32 - 544320*a^28 + 362880*a^24)*w^4 + ((-90720*a^4 + 181440)/a^4)*w^3 + ((-90720*a^16 + 544320*a^12 - 952560*a^8 + 725760*a^4 - 181440)/a^24)*w^2,\n", " 0,\n", " (-332640*a^70 + 3326400*a^54 + 6652800*a^46 - 19958400*a^42 - 29937600*a^38 + 79833600*a^34 - 39916800*a^30)*w^5 + ((-3326400*a^12 + 19958400*a^4 - 19958400)/a^6)*w^4 + ((9979200*a^16 - 59875200*a^12 + 104781600*a^8 - 79833600*a^4 + 19958400)/a^34)*w^3,\n", " 0,\n", " (-8648640*a^96 + 103783680*a^76 + 259459200*a^64 - 605404800*a^60 - 3113510400*a^52 + 3372969600*a^48 + 9340531200*a^44 - 15567552000*a^40 + 6227020800*a^36)*w^6 + ((-129729600*a^24 + 1037836800*a^12 + 778377600*a^8 - 4670265600*a^4 + 3113510400)/a^8)*w^5 + ((778377600*a^20 - 6227020800*a^16 + 17513496000*a^12 - 22572950400*a^8 + 14010796800*a^4 - 3113510400)/a^44)*w^4 + ((-1037836800*a^48 + 8821612800*a^36 + 6227020800*a^32 - 40475635200*a^28 + 23870246400*a^24 + 778377600*a^20 + 23351328000*a^16 - 19848628800*a^12 - 18681062400*a^8 + 23351328000*a^4 - 6227020800)/a^60)*w^3,\n", " 0]" ] }, "execution_count": 223, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# What if we transform this CGF ?\n", "# CGF of contradictory strongly connected implication digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "q2 = exp_had_prod(PhiTwoFourTwo,z*q1.subs(z = z/a),N)\n", "Q2 = [q2[i] * i.factorial() for i in srange(N)]\n", "Q2" ] }, { "cell_type": "code", "execution_count": 225, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " 1,\n", " 0,\n", " -a^12*w,\n", " 0,\n", " (-1/2*a^36 + a^32)*w^2 + 1/2*a^20*w,\n", " 0,\n", " (-1/6*a^72 + a^64 - a^60)*w^3 - 1/2*a^40*w^2,\n", " 0,\n", " (-1/24*a^120 + 1/3*a^108 + 1/4*a^104 - 3/2*a^100 + a^96)*w^4 + (-1/4*a^72 + 1/2*a^68)*w^3 + (-1/4*a^64 + 3/2*a^60 - 21/8*a^56 + 2*a^52 - 1/2*a^48)*w^2,\n", " 0,\n", " (-1/120*a^180 + 1/12*a^164 + 1/6*a^156 - 1/2*a^152 - 3/4*a^148 + 2*a^144 - a^140)*w^5 + (-1/12*a^116 + 1/2*a^108 - 1/2*a^104)*w^4 + (1/4*a^92 - 3/2*a^88 + 21/8*a^84 - 2*a^80 + 1/2*a^76)*w^3,\n", " 0,\n", " (-1/720*a^252 + 1/60*a^232 + 1/24*a^220 - 7/72*a^216 - 1/2*a^208 + 13/24*a^204 + 3/2*a^200 - 5/2*a^196 + a^192)*w^6 + (-1/48*a^172 + 1/6*a^160 + 1/8*a^156 - 3/4*a^152 + 1/2*a^148)*w^5 + (1/8*a^132 - a^128 + 45/16*a^124 - 29/8*a^120 + 9/4*a^116 - 1/2*a^112)*w^4 + (-1/6*a^144 + 17/12*a^132 + a^128 - 13/2*a^124 + 23/6*a^120 + 1/8*a^116 + 15/4*a^112 - 51/16*a^108 - 3*a^104 + 15/4*a^100 - a^96)*w^3,\n", " 0]" ] }, "execution_count": 225, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of contradictory strongly connected implication digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "Q2zero = [q2[i] * a^(i*(i-1)) for i in srange(N)]\n", "Q2zero" ] }, { "cell_type": "code", "execution_count": 226, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -512, -32768/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 4096, 0, -16777216, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, -524288, -33554432/3, -2336462209024/15, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -4278190080,\n", " 0,\n", " -68719476736,\n", " -8725724278030336,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]" ] }, "execution_count": 226, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of contradictory strongly connected implication digraphs (Table 23)\n", "Q2zeroMatrix = [[Q2zero[i][j].subs(a=sqrt(sqrt(2))) for j in srange(N)] for i in srange(N)]\n", "Q2zeroMatrix" ] }, { "cell_type": "code", "execution_count": 227, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " 1,\n", " 0,\n", " -6*a^6*w,\n", " 0,\n", " (-60*a^16 + 120*a^12)*w^2 + 60*w,\n", " 0,\n", " (-840*a^30 + 5040*a^22 - 5040*a^18)*w^3 - 2520/a^2*w^2,\n", " 0,\n", " (-15120*a^48 + 120960*a^36 + 90720*a^32 - 544320*a^28 + 362880*a^24)*w^4 + ((-90720*a^4 + 181440)/a^4)*w^3 + ((-90720*a^16 + 544320*a^12 - 952560*a^8 + 725760*a^4 - 181440)/a^24)*w^2,\n", " 0,\n", " (-332640*a^70 + 3326400*a^54 + 6652800*a^46 - 19958400*a^42 - 29937600*a^38 + 79833600*a^34 - 39916800*a^30)*w^5 + ((-3326400*a^12 + 19958400*a^4 - 19958400)/a^6)*w^4 + ((9979200*a^16 - 59875200*a^12 + 104781600*a^8 - 79833600*a^4 + 19958400)/a^34)*w^3,\n", " 0,\n", " (-8648640*a^96 + 103783680*a^76 + 259459200*a^64 - 605404800*a^60 - 3113510400*a^52 + 3372969600*a^48 + 9340531200*a^44 - 15567552000*a^40 + 6227020800*a^36)*w^6 + ((-129729600*a^24 + 1037836800*a^12 + 778377600*a^8 - 4670265600*a^4 + 3113510400)/a^8)*w^5 + ((778377600*a^20 - 6227020800*a^16 + 17513496000*a^12 - 22572950400*a^8 + 14010796800*a^4 - 3113510400)/a^44)*w^4 + ((-1037836800*a^48 + 8821612800*a^36 + 6227020800*a^32 - 40475635200*a^28 + 23870246400*a^24 + 778377600*a^20 + 23351328000*a^16 - 19848628800*a^12 - 18681062400*a^8 + 23351328000*a^4 - 6227020800)/a^60)*w^3,\n", " 0]" ] }, "execution_count": 227, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# CGF of contradictory strongly connected implication digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "# Counted in a more general way\n", "qq3 = exp(scd.subs(z = a^(10)*z^4*w)/2 - cscc.subs(z = a^(6)*z^4*w)) * exp_had_prod(PhiTwoTwoFour,1/g.subs(z = a^(8)*z^2*w),N)\n", "q3 = exp_had_prod(PhiTwoFourTwo,z*qq3,N)\n", "Q3 = [q3[i] * i.factorial() for i in srange(N)]\n", "Q3" ] }, { "cell_type": "code", "execution_count": 228, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " 1,\n", " 0,\n", " -a^12*w,\n", " 0,\n", " (-1/2*a^36 + a^32)*w^2 + 1/2*a^20*w,\n", " 0,\n", " (-1/6*a^72 + a^64 - a^60)*w^3 - 1/2*a^40*w^2,\n", " 0,\n", " (-1/24*a^120 + 1/3*a^108 + 1/4*a^104 - 3/2*a^100 + a^96)*w^4 + (-1/4*a^72 + 1/2*a^68)*w^3 + (-1/4*a^64 + 3/2*a^60 - 21/8*a^56 + 2*a^52 - 1/2*a^48)*w^2,\n", " 0,\n", " (-1/120*a^180 + 1/12*a^164 + 1/6*a^156 - 1/2*a^152 - 3/4*a^148 + 2*a^144 - a^140)*w^5 + (-1/12*a^116 + 1/2*a^108 - 1/2*a^104)*w^4 + (1/4*a^92 - 3/2*a^88 + 21/8*a^84 - 2*a^80 + 1/2*a^76)*w^3,\n", " 0,\n", " (-1/720*a^252 + 1/60*a^232 + 1/24*a^220 - 7/72*a^216 - 1/2*a^208 + 13/24*a^204 + 3/2*a^200 - 5/2*a^196 + a^192)*w^6 + (-1/48*a^172 + 1/6*a^160 + 1/8*a^156 - 3/4*a^152 + 1/2*a^148)*w^5 + (1/8*a^132 - a^128 + 45/16*a^124 - 29/8*a^120 + 9/4*a^116 - 1/2*a^112)*w^4 + (-1/6*a^144 + 17/12*a^132 + a^128 - 13/2*a^124 + 23/6*a^120 + 1/8*a^116 + 15/4*a^112 - 51/16*a^108 - 3*a^104 + 15/4*a^100 - a^96)*w^3,\n", " 0]" ] }, "execution_count": 228, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of contradictory strongly connected implication digraphs (we take a parameter a = sqrt{sqrt{2}})\n", "Q3zero = [q3[i] * a^(i*(i-1)) for i in srange(N)]\n", "Q3zero" ] }, { "cell_type": "code", "execution_count": 229, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -512, -32768/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 4096, 0, -16777216, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, -524288, -33554432/3, -2336462209024/15, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -4278190080,\n", " 0,\n", " -68719476736,\n", " -8725724278030336,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]" ] }, "execution_count": 229, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of contradictory strongly connected implication digraphs (Table 23)\n", "Q3zeroMatrix = [[Q3zero[i][j].subs(a=sqrt(sqrt(2))) for j in srange(N)] for i in srange(N)]\n", "Q3zeroMatrix" ] }, { "cell_type": "code", "execution_count": 230, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " s,\n", " 0,\n", " -6*a^6*s*w,\n", " 0,\n", " ((-60*a^16 + 120*a^12)*s)*w^2 + 60*s*w,\n", " 0,\n", " ((-840*a^30 + 5040*a^22 - 5040*a^18)*s)*w^3 - 2520/a^2*s*w^2,\n", " 0,\n", " ((-15120*a^48 + 120960*a^36 + 90720*a^32 - 544320*a^28 + 362880*a^24)*s)*w^4 + (((-90720*a^4 + 181440)/a^4)*s)*w^3 + (((181440*a^16 - 725760*a^12 + 1088640*a^8 - 725760*a^4 + 181440)/a^24)*s^2 + ((-90720*a^16 + 544320*a^12 - 952560*a^8 + 725760*a^4 - 181440)/a^24)*s)*w^2,\n", " 0,\n", " ((-332640*a^70 + 3326400*a^54 + 6652800*a^46 - 19958400*a^42 - 29937600*a^38 + 79833600*a^34 - 39916800*a^30)*s)*w^5 + (((-3326400*a^12 + 19958400*a^4 - 19958400)/a^6)*s)*w^4 + (((-19958400*a^16 + 79833600*a^12 - 119750400*a^8 + 79833600*a^4 - 19958400)/a^34)*s^2 + ((9979200*a^16 - 59875200*a^12 + 104781600*a^8 - 79833600*a^4 + 19958400)/a^34)*s)*w^3,\n", " 0,\n", " ((-8648640*a^96 + 103783680*a^76 + 259459200*a^64 - 605404800*a^60 - 3113510400*a^52 + 3372969600*a^48 + 9340531200*a^44 - 15567552000*a^40 + 6227020800*a^36)*s)*w^6 + (((-129729600*a^24 + 1037836800*a^12 + 778377600*a^8 - 4670265600*a^4 + 3113510400)/a^8)*s)*w^5 + (((-1556755200*a^20 + 9340531200*a^16 - 21794572800*a^12 + 24908083200*a^8 - 14010796800*a^4 + 3113510400)/a^44)*s^2 + ((778377600*a^20 - 6227020800*a^16 + 17513496000*a^12 - 22572950400*a^8 + 14010796800*a^4 - 3113510400)/a^44)*s)*w^4 + (((1037836800*a^48 - 8302694400*a^36 - 6227020800*a^32 + 37362124800*a^28 - 20756736000*a^24 + 1556755200*a^20 - 28021593600*a^16 + 21794572800*a^12 + 18681062400*a^8 - 23351328000*a^4 + 6227020800)/a^60)*s^2 + ((-1037836800*a^48 + 8821612800*a^36 + 6227020800*a^32 - 40475635200*a^28 + 23870246400*a^24 + 778377600*a^20 + 23351328000*a^16 - 19848628800*a^12 - 18681062400*a^8 + 23351328000*a^4 - 6227020800)/a^60)*s)*w^3,\n", " 0]" ] }, "execution_count": 230, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# CGF of 2-CNFs, s marks contradictory strongly connected components (we take a parameter a = sqrt{sqrt{2}})\n", "qq4 = exp(scd.subs(z = a^(10)*z^4*w)/2 + (s-1) * cscc.subs(z = a^(6)*z^4*w)) * exp_had_prod(PhiTwoTwoFour,1/g.subs(z = a^(8)*z^2*w),N)\n", "q4 = s * exp_had_prod(PhiTwoFourTwo,z*qq4,N)\n", "Q4 = [q4[i] * i.factorial() for i in srange(N)]\n", "Q4" ] }, { "cell_type": "code", "execution_count": 231, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " s,\n", " 0,\n", " -a^12*s*w,\n", " 0,\n", " ((-1/2*a^36 + a^32)*s)*w^2 + 1/2*a^20*s*w,\n", " 0,\n", " ((-1/6*a^72 + a^64 - a^60)*s)*w^3 - 1/2*a^40*s*w^2,\n", " 0,\n", " ((-1/24*a^120 + 1/3*a^108 + 1/4*a^104 - 3/2*a^100 + a^96)*s)*w^4 + ((-1/4*a^72 + 1/2*a^68)*s)*w^3 + ((1/2*a^64 - 2*a^60 + 3*a^56 - 2*a^52 + 1/2*a^48)*s^2 + (-1/4*a^64 + 3/2*a^60 - 21/8*a^56 + 2*a^52 - 1/2*a^48)*s)*w^2,\n", " 0,\n", " ((-1/120*a^180 + 1/12*a^164 + 1/6*a^156 - 1/2*a^152 - 3/4*a^148 + 2*a^144 - a^140)*s)*w^5 + ((-1/12*a^116 + 1/2*a^108 - 1/2*a^104)*s)*w^4 + ((-1/2*a^92 + 2*a^88 - 3*a^84 + 2*a^80 - 1/2*a^76)*s^2 + (1/4*a^92 - 3/2*a^88 + 21/8*a^84 - 2*a^80 + 1/2*a^76)*s)*w^3,\n", " 0,\n", " ((-1/720*a^252 + 1/60*a^232 + 1/24*a^220 - 7/72*a^216 - 1/2*a^208 + 13/24*a^204 + 3/2*a^200 - 5/2*a^196 + a^192)*s)*w^6 + ((-1/48*a^172 + 1/6*a^160 + 1/8*a^156 - 3/4*a^152 + 1/2*a^148)*s)*w^5 + ((-1/4*a^132 + 3/2*a^128 - 7/2*a^124 + 4*a^120 - 9/4*a^116 + 1/2*a^112)*s^2 + (1/8*a^132 - a^128 + 45/16*a^124 - 29/8*a^120 + 9/4*a^116 - 1/2*a^112)*s)*w^4 + ((1/6*a^144 - 4/3*a^132 - a^128 + 6*a^124 - 10/3*a^120 + 1/4*a^116 - 9/2*a^112 + 7/2*a^108 + 3*a^104 - 15/4*a^100 + a^96)*s^2 + (-1/6*a^144 + 17/12*a^132 + a^128 - 13/2*a^124 + 23/6*a^120 + 1/8*a^116 + 15/4*a^112 - 51/16*a^108 - 3*a^104 + 15/4*a^100 - a^96)*s)*w^3,\n", " 0]" ] }, "execution_count": 231, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNFs, s marks contradictory strongly connected components (we take a parameter a = sqrt{sqrt{2}})\n", "Q4zero = [q4[i] * a^(i*(i-1)) for i in srange(N)]\n", "Q4zero" ] }, { "cell_type": "code", "execution_count": 232, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -8*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 16*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -512*s, -32768/3*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 2048*s^2 + 4096*s, 0, -16777216*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -262144*s^2 - 524288*s,\n", " -33554432/3*s,\n", " -2336462209024/15*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 13497270272/3*s^2 - 4278190080*s,\n", " 0,\n", " -68719476736*s,\n", " -8725724278030336*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]" ] }, "execution_count": 232, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNFs, s marks contradictory strongly connected components (Table 25)\n", "Q4zeroMatrix = [[Q4zero[i][j].subs(a=sqrt(sqrt(2))) for j in srange(N)] for i in srange(N)]\n", "Q4zeroMatrix" ] }, { "cell_type": "code", "execution_count": 234, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -512, -32768/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 4096, 0, -16777216, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, -524288, -33554432/3, -2336462209024/15, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -4278190080,\n", " 0,\n", " -68719476736,\n", " -8725724278030336,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]" ] }, "execution_count": 234, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNFs, s marks contradictory strongly connected components (Table 23)\n", "# We extract [s] to obtain the asymptotics of contradictory strongly connected implication digraphs\n", "Q4zeroMatrixZeroDifS = [[Q4zeroMatrix[i][j].diff(s).subs(s=0) for j in srange(N)] for i in srange(N)]\n", "Q4zeroMatrixZeroDifS" ] }, { "cell_type": "code", "execution_count": 235, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]" ] }, "execution_count": 235, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Verification\n", "QCNFver14 = [[Q1zeroMatrix[i][j] - Q4zeroMatrixZeroDifS[i+1][j] for j in srange(N)] for i in srange(N-1)]\n", "QCNFver14" ] }, { "cell_type": "code", "execution_count": 236, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " s,\n", " 0,\n", " (6*a^6*s*t - 6*a^6*s)*w,\n", " 0,\n", " (((120*a^12 - 60*a^8)*s)*t^2 + ((60*a^16 - 240*a^12 + 60*a^8)*s)*t + (-60*a^16 + 120*a^12)*s)*w^2 + (-60*s*t + 60*s)*w,\n", " 0,\n", " (((5040*a^18 - 5040*a^14 + 840*a^6)*s)*t^3 + ((5040*a^22 - 15120*a^18 + 5040*a^14 + 5040*a^10 - 2520*a^6)*s)*t^2 + ((840*a^30 - 10080*a^22 + 15120*a^18 - 5040*a^10 + 1680*a^6)*s)*t + (-840*a^30 + 5040*a^22 - 5040*a^18)*s)*w^3 + (-2520*a^6*s*t^2 + (((2520*a^8 + 2520)/a^2)*s)*t - 2520/a^2*s)*w^2,\n", " 0,\n", " (((362880*a^24 - 544320*a^20 + 90720*a^16 + 120960*a^12 - 15120)*s)*t^4 + ((544320*a^28 - 1632960*a^24 + 907200*a^20 + 544320*a^16 - 423360*a^12 + 90720*a^8 - 181440*a^4 + 90720)*s)*t^3 + ((120960*a^36 + 90720*a^32 - 1632960*a^28 + 2388960*a^24 - 362880*a^20 - 680400*a^16 + 241920*a^12 - 453600*a^8 + 544320*a^4 - 166320)*s)*t^2 + ((15120*a^48 - 241920*a^36 - 181440*a^32 + 1632960*a^28 - 1481760*a^24 + 45360*a^16 + 60480*a^12 + 362880*a^8 - 362880*a^4 + 90720)*s)*t + (-15120*a^48 + 120960*a^36 + 90720*a^32 - 544320*a^28 + 362880*a^24)*s)*w^4 + (((-181440*a^12 + 90720*a^8)*s)*t^3 + ((-90720*a^16 + 362880*a^12 - 181440*a^8 + 181440*a^4)*s)*t^2 + (((90720*a^20 - 181440*a^16 + 90720*a^12 - 181440*a^8 + 90720*a^4 - 181440)/a^4)*s)*t + ((-90720*a^4 + 181440)/a^4)*s)*w^3 + (45360/a^16*s*t^2 + (((-90720*a^8 + 181440*a^4 - 181440)/a^16)*s)*t + ((181440*a^16 - 725760*a^12 + 1088640*a^8 - 725760*a^4 + 181440)/a^24)*s^2 + ((-90720*a^16 + 544320*a^12 - 952560*a^8 + 725760*a^4 - 181440)/a^24)*s)*w^2,\n", " 0,\n", " ((((39916800*a^40 - 79833600*a^36 + 29937600*a^32 + 19958400*a^28 - 6652800*a^24 - 3326400*a^16 + 332640)/a^10)*s)*t^5 + (((79833600*a^44 - 259459200*a^40 + 199584000*a^36 + 76507200*a^32 - 126403200*a^28 + 46569600*a^24 - 39916800*a^20 + 21621600*a^16 - 3326400*a^8 + 6652800*a^4 - 3326400)/a^10)*s)*t^4 + (((19958400*a^52 + 23284800*a^48 - 319334400*a^44 + 505612800*a^40 - 119750400*a^36 - 222868800*a^32 + 166320000*a^28 - 139708800*a^24 + 139708800*a^20 - 61538400*a^16 + 39916800*a^8 - 39916800*a^4 + 11642400)/a^10)*s)*t^3 + (((3326400*a^64 + 6652800*a^56 - 59875200*a^52 - 81496800*a^48 + 479001600*a^44 - 449064000*a^40 - 6652800*a^36 + 134719200*a^32 - 106444800*a^28 + 192931200*a^24 - 199584000*a^20 + 123076800*a^16 - 96465600*a^8 + 73180800*a^4 - 16632000)/a^10)*s)*t^2 + (((332640*a^80 - 6652800*a^64 - 13305600*a^56 + 59875200*a^52 + 88149600*a^48 - 319334400*a^44 + 202910400*a^40 + 6652800*a^36 - 18295200*a^32 + 46569600*a^28 - 93139200*a^24 + 99792000*a^20 - 79833600*a^16 + 59875200*a^8 - 39916800*a^4 + 7983360)/a^10)*s)*t + (-332640*a^70 + 3326400*a^54 + 6652800*a^46 - 19958400*a^42 - 29937600*a^38 + 79833600*a^34 - 39916800*a^30)*s)*w^5 + (((-19958400*a^18 + 19958400*a^14 - 3326400*a^6)*s)*t^4 + ((-19958400*a^22 + 59875200*a^18 - 29937600*a^14 + 3326400*a^6)*s)*t^3 + ((-3326400*a^30 + 39916800*a^22 - 59875200*a^18 + 9979200*a^14 + 13305600*a^6 - 19958400*a^2)*s)*t^2 + (((3326400*a^36 - 19958400*a^28 + 19958400*a^24 - 9979200*a^12 + 19958400*a^8 - 19958400*a^4 + 19958400)/a^6)*s)*t + ((-3326400*a^12 + 19958400*a^4 - 19958400)/a^6)*s)*w^4 + (4989600/a^10*s*t^3 + (((-9979200*a^24 + 19958400*a^20 - 19958400*a^16 - 4989600)/a^26)*s)*t^2 + (((19958400*a^16 - 79833600*a^12 + 119750400*a^8 - 79833600*a^4 + 19958400)/a^18)*s^2 + ((-9979200*a^24 + 59875200*a^20 - 104781600*a^16 + 79833600*a^12 - 9979200*a^8 - 19958400*a^4 + 19958400)/a^26)*s)*t + ((-19958400*a^16 + 79833600*a^12 - 119750400*a^8 + 79833600*a^4 - 19958400)/a^34)*s^2 + ((9979200*a^16 - 59875200*a^12 + 104781600*a^8 - 79833600*a^4 + 19958400)/a^34)*s)*w^3,\n", " 0,\n", " ((((6227020800*a^60 - 15567552000*a^56 + 9340531200*a^52 + 3372969600*a^48 - 3113510400*a^44 - 605404800*a^36 + 259459200*a^32 + 103783680*a^20 - 8648640)/a^24)*s)*t^6 + (((15567552000*a^64 - 56043187200*a^60 + 55264809600*a^56 + 9340531200*a^52 - 38140502400*a^48 + 17124307200*a^44 - 9081072000*a^40 + 7783776000*a^36 - 1816214400*a^32 - 1037836800*a^28 + 2075673600*a^24 - 1089728640*a^20 + 129729600*a^8 - 259459200*a^4 + 129729600)/a^24)*s)*t^5 + (((4151347200*a^72 + 6227020800*a^68 - 80172892800*a^64 + 142010668800*a^60 - 51113462400*a^56 - 68756688000*a^52 + 64345881600*a^48 - 37881043200*a^44 + 43199956800*a^40 - 29924294400*a^36 + 4410806400*a^32 + 12972960000*a^28 - 13664851200*a^24 + 4151347200*a^20 + 648648000*a^16 + 518918400*a^12 - 2854051200*a^8 + 2594592000*a^4 - 735134400)/a^24)*s)*t^4 + (((778377600*a^84 - 259459200*a^80 + 3113510400*a^76 - 15827011200*a^72 - 31654022400*a^68 + 162162000000*a^64 - 166745779200*a^60 + 2075673600*a^56 + 73945872000*a^52 - 48648600000*a^48 + 64864800000*a^44 - 85102617600*a^40 + 58897238400*a^36 - 4670265600*a^32 - 32691859200*a^28 + 23999976000*a^24 + 1556755200*a^20 - 8562153600*a^16 - 4151347200*a^12 + 13881067200*a^8 - 9081072000*a^4 + 1945944000)/a^24)*s)*t^3 + (((103783680*a^100 + 259459200*a^88 - 2162160000*a^84 + 337296960*a^80 - 9340531200*a^76 + 22572950400*a^72 + 53189136000*a^68 - 160605244800*a^64 + 106551244800*a^60 + 11675664000*a^56 - 29059430400*a^52 + 24432408000*a^48 - 45145900800*a^44 + 60843182400*a^40 - 53794540800*a^36 + 11675664000*a^32 + 22832409600*a^28 - 10594584000*a^24 - 20601060480*a^20 + 21145924800*a^16 + 8821612800*a^12 - 23610787200*a^8 + 12972960000*a^4 - 2369727360)/a^24)*s)*t^2 + (((8648640*a^120 - 207567360*a^100 - 518918400*a^88 + 1989187200*a^84 - 77837760*a^80 + 9340531200*a^76 - 14270256000*a^72 - 37102665600*a^68 + 78616137600*a^64 - 38226988800*a^60 - 2335132800*a^56 + 5189184000*a^52 - 5362156800*a^48 + 4151347200*a^44 - 9859449600*a^40 + 17643225600*a^36 - 9859449600*a^32 - 2075673600*a^28 - 1816214400*a^24 + 15878903040*a^20 - 13232419200*a^16 - 5189184000*a^12 + 12454041600*a^8 - 6227020800*a^4 + 1037836800)/a^24)*s)*t + (-8648640*a^96 + 103783680*a^76 + 259459200*a^64 - 605404800*a^60 - 3113510400*a^52 + 3372969600*a^48 + 9340531200*a^44 - 15567552000*a^40 + 6227020800*a^36)*s)*w^6 + (((-3113510400*a^24 + 4670265600*a^20 - 778377600*a^16 - 1037836800*a^12 + 129729600)*s)*t^5 + ((-4670265600*a^28 + 14010796800*a^24 - 9340531200*a^20 - 778377600*a^16 + 1037836800*a^12 - 778377600*a^8 + 2075673600*a^4 - 908107200)*s)*t^4 + ((-1037836800*a^36 - 778377600*a^32 + 14010796800*a^28 - 21275654400*a^24 + 7783776000*a^20 - 1945944000*a^16 + 2594592000*a^12 + 3891888000*a^8 - 6745939200*a^4 + 2205403200)*s)*t^3 + ((-129729600*a^48 + 2075673600*a^36 + 1556755200*a^32 - 13491878400*a^28 + 12972960000*a^24 - 6227020800*a^20 + 10118908800*a^16 - 5189184000*a^12 - 5448643200*a^8 + 4151347200*a^4 + 908107200)*s)*t^2 + (((129729600*a^56 - 1037836800*a^44 - 778377600*a^40 + 4151347200*a^36 - 2594592000*a^32 + 3113510400*a^28 - 6486480000*a^24 + 2594592000*a^20 + 2335132800*a^16 - 518918400*a^12 - 3113510400*a^8 + 4670265600*a^4 - 3113510400)/a^8)*s)*t + ((-129729600*a^24 + 1037836800*a^12 + 778377600*a^8 - 4670265600*a^4 + 3113510400)/a^8)*s)*w^5 + ((((778377600*a^4 - 389188800)/a^8)*s)*t^4 + (((-1556755200*a^24 + 4281076800*a^20 - 5448643200*a^16 + 1945944000*a^12 - 778377600)/a^20)*s)*t^3 + (((3113510400*a^20 - 14010796800*a^16 + 24908083200*a^12 - 21794572800*a^8 + 9340531200*a^4 - 1556755200)/a^16)*s^2 + ((-778377600*a^44 + 1556755200*a^40 + 4670265600*a^36 - 16345929600*a^32 + 19070251200*a^28 - 7783776000*a^24 - 1556755200*a^20 + 3113510400*a^16 - 389188800*a^4 + 778377600)/a^36)*s)*t^2 + (((1556755200*a^36 - 9340531200*a^32 + 23351328000*a^28 - 31135104000*a^24 + 23351328000*a^20 - 12454041600*a^16 + 14010796800*a^12 - 18681062400*a^8 + 12454041600*a^4 - 3113510400)/a^28)*s^2 + ((-778377600*a^44 + 6227020800*a^40 - 18291873600*a^36 + 27243216000*a^32 - 22183761600*a^28 + 10897286400*a^24 - 10897286400*a^20 + 16345929600*a^16 - 11675664000*a^12 + 4670265600*a^4 - 3113510400)/a^36)*s)*t + ((-1556755200*a^20 + 9340531200*a^16 - 21794572800*a^12 + 24908083200*a^8 - 14010796800*a^4 + 3113510400)/a^44)*s^2 + ((778377600*a^20 - 6227020800*a^16 + 17513496000*a^12 - 22572950400*a^8 + 14010796800*a^4 - 3113510400)/a^44)*s)*w^4 + (-129729600/a^48*s*t^3 + (((778377600*a^8 - 1556755200*a^4 + 1167566400)/a^48)*s)*t^2 + (((-1556755200*a^16 + 6227020800*a^12 - 9340531200*a^8 + 6227020800*a^4 - 1556755200)/a^56)*s^2 + ((-518918400*a^32 + 3113510400*a^24 - 3113510400*a^20 - 1556755200*a^16 + 6356750400*a^8 - 6227020800*a^4 + 1556755200)/a^56)*s)*t + ((1037836800*a^48 - 8302694400*a^36 - 6227020800*a^32 + 37362124800*a^28 - 20756736000*a^24 + 1556755200*a^20 - 28021593600*a^16 + 21794572800*a^12 + 18681062400*a^8 - 23351328000*a^4 + 6227020800)/a^60)*s^2 + ((-1037836800*a^48 + 8821612800*a^36 + 6227020800*a^32 - 40475635200*a^28 + 23870246400*a^24 + 778377600*a^20 + 23351328000*a^16 - 19848628800*a^12 - 18681062400*a^8 + 23351328000*a^4 - 6227020800)/a^60)*s)*w^3,\n", " 0]" ] }, "execution_count": 236, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# CGF of 2-CNFs (we take a parameter a = sqrt{sqrt{2}})\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "qq5 = exp((1-t) * scd.subs(z = a^(10)*z^4*w)/2 + (s-1) * cscc.subs(z = a^(6)*z^4*w)) * exp_had_prod(PhiTwoTwoFour,1/g.subs(z = a^(8)*z^2*w),N)\n", "q5 = s * dt.subs(z = a^6*z^2*w) * exp_had_prod(PhiTwoFourTwo,z*qq5,N)\n", "Q5 = [q5[i] * i.factorial() for i in srange(N)]\n", "Q5" ] }, { "cell_type": "code", "execution_count": 237, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0,\n", " s,\n", " 0,\n", " (a^12*s*t - a^12*s)*w,\n", " 0,\n", " (((a^32 - 1/2*a^28)*s)*t^2 + ((1/2*a^36 - 2*a^32 + 1/2*a^28)*s)*t + (-1/2*a^36 + a^32)*s)*w^2 + (-1/2*a^20*s*t + 1/2*a^20*s)*w,\n", " 0,\n", " (((a^60 - a^56 + 1/6*a^48)*s)*t^3 + ((a^64 - 3*a^60 + a^56 + a^52 - 1/2*a^48)*s)*t^2 + ((1/6*a^72 - 2*a^64 + 3*a^60 - a^52 + 1/3*a^48)*s)*t + (-1/6*a^72 + a^64 - a^60)*s)*w^3 + (-1/2*a^48*s*t^2 + ((1/2*a^48 + 1/2*a^40)*s)*t - 1/2*a^40*s)*w^2,\n", " 0,\n", " (((a^96 - 3/2*a^92 + 1/4*a^88 + 1/3*a^84 - 1/24*a^72)*s)*t^4 + ((3/2*a^100 - 9/2*a^96 + 5/2*a^92 + 3/2*a^88 - 7/6*a^84 + 1/4*a^80 - 1/2*a^76 + 1/4*a^72)*s)*t^3 + ((1/3*a^108 + 1/4*a^104 - 9/2*a^100 + 79/12*a^96 - a^92 - 15/8*a^88 + 2/3*a^84 - 5/4*a^80 + 3/2*a^76 - 11/24*a^72)*s)*t^2 + ((1/24*a^120 - 2/3*a^108 - 1/2*a^104 + 9/2*a^100 - 49/12*a^96 + 1/8*a^88 + 1/6*a^84 + a^80 - a^76 + 1/4*a^72)*s)*t + (-1/24*a^120 + 1/3*a^108 + 1/4*a^104 - 3/2*a^100 + a^96)*s)*w^4 + (((-1/2*a^84 + 1/4*a^80)*s)*t^3 + ((-1/4*a^88 + a^84 - 1/2*a^80 + 1/2*a^76)*s)*t^2 + ((1/4*a^88 - 1/2*a^84 + 1/4*a^80 - 1/2*a^76 + 1/4*a^72 - 1/2*a^68)*s)*t + (-1/4*a^72 + 1/2*a^68)*s)*w^3 + (1/8*a^56*s*t^2 + ((-1/4*a^64 + 1/2*a^60 - 1/2*a^56)*s)*t + (1/2*a^64 - 2*a^60 + 3*a^56 - 2*a^52 + 1/2*a^48)*s^2 + (-1/4*a^64 + 3/2*a^60 - 21/8*a^56 + 2*a^52 - 1/2*a^48)*s)*w^2,\n", " 0,\n", " (((a^140 - 2*a^136 + 3/4*a^132 + 1/2*a^128 - 1/6*a^124 - 1/12*a^116 + 1/120*a^100)*s)*t^5 + ((2*a^144 - 13/2*a^140 + 5*a^136 + 23/12*a^132 - 19/6*a^128 + 7/6*a^124 - a^120 + 13/24*a^116 - 1/12*a^108 + 1/6*a^104 - 1/12*a^100)*s)*t^4 + ((1/2*a^152 + 7/12*a^148 - 8*a^144 + 38/3*a^140 - 3*a^136 - 67/12*a^132 + 25/6*a^128 - 7/2*a^124 + 7/2*a^120 - 37/24*a^116 + a^108 - a^104 + 7/24*a^100)*s)*t^3 + ((1/12*a^164 + 1/6*a^156 - 3/2*a^152 - 49/24*a^148 + 12*a^144 - 45/4*a^140 - 1/6*a^136 + 27/8*a^132 - 8/3*a^128 + 29/6*a^124 - 5*a^120 + 37/12*a^116 - 29/12*a^108 + 11/6*a^104 - 5/12*a^100)*s)*t^2 + ((1/120*a^180 - 1/6*a^164 - 1/3*a^156 + 3/2*a^152 + 53/24*a^148 - 8*a^144 + 61/12*a^140 + 1/6*a^136 - 11/24*a^132 + 7/6*a^128 - 7/3*a^124 + 5/2*a^120 - 2*a^116 + 3/2*a^108 - a^104 + 1/5*a^100)*s)*t + (-1/120*a^180 + 1/12*a^164 + 1/6*a^156 - 1/2*a^152 - 3/4*a^148 + 2*a^144 - a^140)*s)*w^5 + (((-1/2*a^128 + 1/2*a^124 - 1/12*a^116)*s)*t^4 + ((-1/2*a^132 + 3/2*a^128 - 3/4*a^124 + 1/12*a^116)*s)*t^3 + ((-1/12*a^140 + a^132 - 3/2*a^128 + 1/4*a^124 + 1/3*a^116 - 1/2*a^112)*s)*t^2 + ((1/12*a^140 - 1/2*a^132 + 1/2*a^128 - 1/4*a^116 + 1/2*a^112 - 1/2*a^108 + 1/2*a^104)*s)*t + (-1/12*a^116 + 1/2*a^108 - 1/2*a^104)*s)*w^4 + (1/8*a^100*s*t^3 + ((-1/4*a^108 + 1/2*a^104 - 1/2*a^100 - 1/8*a^84)*s)*t^2 + ((1/2*a^108 - 2*a^104 + 3*a^100 - 2*a^96 + 1/2*a^92)*s^2 + (-1/4*a^108 + 3/2*a^104 - 21/8*a^100 + 2*a^96 - 1/4*a^92 - 1/2*a^88 + 1/2*a^84)*s)*t + (-1/2*a^92 + 2*a^88 - 3*a^84 + 2*a^80 - 1/2*a^76)*s^2 + (1/4*a^92 - 3/2*a^88 + 21/8*a^84 - 2*a^80 + 1/2*a^76)*s)*w^3,\n", " 0,\n", " (((a^192 - 5/2*a^188 + 3/2*a^184 + 13/24*a^180 - 1/2*a^176 - 7/72*a^168 + 1/24*a^164 + 1/60*a^152 - 1/720*a^132)*s)*t^6 + ((5/2*a^196 - 9*a^192 + 71/8*a^188 + 3/2*a^184 - 49/8*a^180 + 11/4*a^176 - 35/24*a^172 + 5/4*a^168 - 7/24*a^164 - 1/6*a^160 + 1/3*a^156 - 7/40*a^152 + 1/48*a^140 - 1/24*a^136 + 1/48*a^132)*s)*t^5 + ((2/3*a^204 + a^200 - 103/8*a^196 + 821/36*a^192 - 197/24*a^188 - 265/24*a^184 + 31/3*a^180 - 73/12*a^176 + 111/16*a^172 - 173/36*a^168 + 17/24*a^164 + 25/12*a^160 - 79/36*a^156 + 2/3*a^152 + 5/48*a^148 + 1/12*a^144 - 11/24*a^140 + 5/12*a^136 - 17/144*a^132)*s)*t^4 + ((1/8*a^216 - 1/24*a^212 + 1/2*a^208 - 61/24*a^204 - 61/12*a^200 + 625/24*a^196 - 241/9*a^192 + 1/3*a^188 + 95/8*a^184 - 125/16*a^180 + 125/12*a^176 - 41/3*a^172 + 227/24*a^168 - 3/4*a^164 - 21/4*a^160 + 185/48*a^156 + 1/4*a^152 - 11/8*a^148 - 2/3*a^144 + 107/48*a^140 - 35/24*a^136 + 5/16*a^132)*s)*t^3 + ((1/60*a^232 + 1/24*a^220 - 25/72*a^216 + 13/240*a^212 - 3/2*a^208 + 29/8*a^204 + 205/24*a^200 - 619/24*a^196 + 154/9*a^192 + 15/8*a^188 - 14/3*a^184 + 565/144*a^180 - 29/4*a^176 + 469/48*a^172 - 311/36*a^168 + 15/8*a^164 + 11/3*a^160 - 245/144*a^156 - 397/120*a^152 + 163/48*a^148 + 17/12*a^144 - 91/24*a^140 + 25/12*a^136 - 137/360*a^132)*s)*t^2 + ((1/720*a^252 - 1/30*a^232 - 1/12*a^220 + 23/72*a^216 - 1/80*a^212 + 3/2*a^208 - 55/24*a^204 - 143/24*a^200 + 101/8*a^196 - 221/36*a^192 - 3/8*a^188 + 5/6*a^184 - 31/36*a^180 + 2/3*a^176 - 19/12*a^172 + 17/6*a^168 - 19/12*a^164 - 1/3*a^160 - 7/24*a^156 + 51/20*a^152 - 17/8*a^148 - 5/6*a^144 + 2*a^140 - a^136 + 1/6*a^132)*s)*t + (-1/720*a^252 + 1/60*a^232 + 1/24*a^220 - 7/72*a^216 - 1/2*a^208 + 13/24*a^204 + 3/2*a^200 - 5/2*a^196 + a^192)*s)*w^6 + (((-1/2*a^180 + 3/4*a^176 - 1/8*a^172 - 1/6*a^168 + 1/48*a^156)*s)*t^5 + ((-3/4*a^184 + 9/4*a^180 - 3/2*a^176 - 1/8*a^172 + 1/6*a^168 - 1/8*a^164 + 1/3*a^160 - 7/48*a^156)*s)*t^4 + ((-1/6*a^192 - 1/8*a^188 + 9/4*a^184 - 41/12*a^180 + 5/4*a^176 - 5/16*a^172 + 5/12*a^168 + 5/8*a^164 - 13/12*a^160 + 17/48*a^156)*s)*t^3 + ((-1/48*a^204 + 1/3*a^192 + 1/4*a^188 - 13/6*a^184 + 25/12*a^180 - a^176 + 13/8*a^172 - 5/6*a^168 - 7/8*a^164 + 2/3*a^160 + 7/48*a^156)*s)*t^2 + ((1/48*a^204 - 1/6*a^192 - 1/8*a^188 + 2/3*a^184 - 5/12*a^180 + 1/2*a^176 - 25/24*a^172 + 5/12*a^168 + 3/8*a^164 - 1/12*a^160 - 1/2*a^156 + 3/4*a^152 - 1/2*a^148)*s)*t + (-1/48*a^172 + 1/6*a^160 + 1/8*a^156 - 3/4*a^152 + 1/2*a^148)*s)*w^5 + (((1/8*a^152 - 1/16*a^148)*s)*t^4 + ((-1/4*a^160 + 11/16*a^156 - 7/8*a^152 + 5/16*a^148 - 1/8*a^136)*s)*t^3 + ((1/2*a^160 - 9/4*a^156 + 4*a^152 - 7/2*a^148 + 3/2*a^144 - 1/4*a^140)*s^2 + (-1/8*a^164 + 1/4*a^160 + 3/4*a^156 - 21/8*a^152 + 49/16*a^148 - 5/4*a^144 - 1/4*a^140 + 1/2*a^136 - 1/16*a^124 + 1/8*a^120)*s)*t^2 + ((1/4*a^164 - 3/2*a^160 + 15/4*a^156 - 5*a^152 + 15/4*a^148 - 2*a^144 + 9/4*a^140 - 3*a^136 + 2*a^132 - 1/2*a^128)*s^2 + (-1/8*a^164 + a^160 - 47/16*a^156 + 35/8*a^152 - 57/16*a^148 + 7/4*a^144 - 7/4*a^140 + 21/8*a^136 - 15/8*a^132 + 3/4*a^124 - 1/2*a^120)*s)*t + (-1/4*a^132 + 3/2*a^128 - 7/2*a^124 + 4*a^120 - 9/4*a^116 + 1/2*a^112)*s^2 + (1/8*a^132 - a^128 + 45/16*a^124 - 29/8*a^120 + 9/4*a^116 - 1/2*a^112)*s)*w^4 + (-1/48*a^108*s*t^3 + ((1/8*a^116 - 1/4*a^112 + 3/16*a^108)*s)*t^2 + ((-1/4*a^116 + a^112 - 3/2*a^108 + a^104 - 1/4*a^100)*s^2 + (-1/12*a^132 + 1/2*a^124 - 1/2*a^120 - 1/4*a^116 + 49/48*a^108 - a^104 + 1/4*a^100)*s)*t + (1/6*a^144 - 4/3*a^132 - a^128 + 6*a^124 - 10/3*a^120 + 1/4*a^116 - 9/2*a^112 + 7/2*a^108 + 3*a^104 - 15/4*a^100 + a^96)*s^2 + (-1/6*a^144 + 17/12*a^132 + a^128 - 13/2*a^124 + 23/6*a^120 + 1/8*a^116 + 15/4*a^112 - 51/16*a^108 - 3*a^104 + 15/4*a^100 - a^96)*s)*w^3,\n", " 0]" ] }, "execution_count": 237, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNFs (we take a parameter a = sqrt{sqrt{2}})\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "Q5zero = [q5[i] * a^(i*(i-1)) for i in srange(N)]\n", "Q5zero" ] }, { "cell_type": "code", "execution_count": 238, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 8*s*t - 8*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -16*s*t + 16*s, 192*s*t^2 - 192*s*t, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " -2048*s*t^2 + 2560*s*t - 512*s,\n", " 51200/3*s*t^3 - 10240*s*t^2 + 4096*s*t - 32768/3*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 2048*s*t^2 + 2048*s^2 - 8192*s*t + 4096*s,\n", " -786432*s*t^3 + 786432*s*t^2,\n", " 5931008*s*t^4 - 851968/3*s*t^3 + 5472256*s*t^2 + 16973824/3*s*t - 16777216*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 4194304*s*t^3 - 17039360*s*t^2 - 262144*s^2 + 1048576*(4*s^2 + 9*s)*t - 524288*s,\n", " -3355443200/3*s*t^4 + 1744830464/3*s*t^3 - 134217728*s*t^2 + 2046820352/3*s*t - 33554432/3*s,\n", " 122813415424/15*s*t^5 + 13748928512/3*s*t^4 + 16965959680*s*t^3 + 123505475584/3*s*t^2 + 84859158528*s*t - 2336462209024/15*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -8388608/3*s*t^3 + 25165824*s*t^2 + 13497270272/3*s^2 - 8388608*(s^2 + 28*s)*t - 4278190080*s,\n", " 25769803776*s*t^4 - 96636764160*s*t^3 + 25769803776*(s^2 + s)*t^2 + 6442450944*(s^2 + 2*s)*t,\n", " -6219112644608*s*t^5 - 1065151889408/3*s*t^4 - 14809047236608/3*s*t^3 - 15427522527232/3*s*t^2 + 50165218017280/3*s*t - 68719476736*s,\n", " 225575440482304/5*s*t^6 + 803299007660032/15*s*t^5 + 496140126519296/3*s*t^4 + 4602751631753216/9*s*t^3 + 5580082714247168/3*s*t^2 + 274060415753781248/45*s*t - 8725724278030336*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]" ] }, "execution_count": 238, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNFs (Table 24)\n", "# s is the marking variable for contradictory strongly connected components\n", "# t is the marking variable for pairs of ordinary strongly connected components\n", "Q5zeroMatrix = [[Q5zero[i][j].subs(a=sqrt(sqrt(2))) for j in srange(N)] for i in srange(N)]\n", "Q5zeroMatrix" ] }, { "cell_type": "code", "execution_count": 239, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, -8*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 16*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, -512*s, -32768/3*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 2048*s^2 + 4096*s, 0, -16777216*s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " -262144*s^2 - 524288*s,\n", " -33554432/3*s,\n", " -2336462209024/15*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0,\n", " 0,\n", " 0,\n", " 13497270272/3*s^2 - 4278190080*s,\n", " 0,\n", " -68719476736*s,\n", " -8725724278030336*s,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0,\n", " 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]" ] }, "execution_count": 239, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Coefficients of the CGF of 2-CNFs (Table 25)\n", "# t=0 is the marking variable for pairs of ordinary strongly connected components (no ordinary components)\n", "# s is the marking variable for contradictory strongly connected components\n", "# (we extract [s] to obtain the asymptotics of contradictory strongly connected implication digraphs)\n", "Q5zeroMatrixZeroT = [[Q5zeroMatrix[i][j].subs(t=0) for j in srange(N)] for i in srange(N)]\n", "Q5zeroMatrixZeroT" ] }, { "cell_type": "code", "execution_count": 240, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]" ] }, "execution_count": 240, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Verification\n", "QCNFver45 = [[Q4zeroMatrix[i][j] - Q5zeroMatrixZeroT[i][j] for j in srange(N)] for i in srange(N)]\n", "QCNFver45" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.0", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 4 }