{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"color:black; font-weight: bold; font-size:26px\">\n",
    "    Labeled digraphs with (strongly) connected components\n",
    "</span>\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 Section A.9, Tables 12-17"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We mark all strongly connected components (by the variable t) and source-like components (by the variable s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"color:black; font-weight: bold; font-size:18px\">\n",
    "    Preliminary section\n",
    "</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 12 # Our accuracy\n",
    "disp = 11 # How many terms we want to display\n",
    "P.<a> = PolynomialRing(QQ)\n",
    "PP.<s> = PolynomialRing(P)\n",
    "PQ.<t> = PolynomialRing(PP)\n",
    "PR.<w> = PolynomialRing(PQ)\n",
    "R.<z> = PowerSeriesRing(PR,N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "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": 5,
   "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, 1/a^90]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Transformation function hat(set)\n",
    "hatset = sum(((1 / a^(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": 6,
   "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, 1/a^45]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Transformation function Phi_{a^2}^{2,1}\n",
    "PhiTwoTwoOne = sum((1 / a^(i*(i-1)/2) / 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": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, a, a^3, a^6, a^10, a^15, a^21, a^28, a^36, a^45]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Transformation function Phi_{a^2}^{1,2}\n",
    "PhiTwoOneTwo = sum((a^(i*(i-1)/2) / i.factorial()) * z^i for i in srange(N))\n",
    "PTOT = [PhiTwoOneTwo[i] * i.factorial() for i in srange(disp)]\n",
    "PTOT"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"color:black; font-weight: bold; font-size:18px\">\n",
    "    Defining supplementary EGFs\n",
    "</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, a^2, a^6, a^12, a^20, a^30, a^42, a^56, a^72, a^90]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# EGF of labeled graphs/tournaments or GGF of labeled digraphs (we take a parameter a = sqrt{2})\n",
    "g = sum((a^(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": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456, 68719476736, 35184372088832]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Numbers of labeled graphs/tournaments\n",
    "Gsub = [g[i][0][0].subs(a=sqrt(2)) * i.factorial() for i in srange(disp)]\n",
    "Gsub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0,\n",
       " 1,\n",
       " a^4 - 2*a^2 + 1,\n",
       " a^12 - 6*a^8 + 6*a^6 + 3*a^4 - 6*a^2 + 2,\n",
       " a^24 - 8*a^18 - 6*a^16 + 36*a^14 - 20*a^12 - 21*a^8 + 12*a^6 + 24*a^4 - 24*a^2 + 6,\n",
       " a^40 - 10*a^32 - 20*a^28 + 60*a^26 + 95*a^24 - 240*a^22 + 120*a^20 - 40*a^18 - 20*a^16 + 160*a^14 - 160*a^12 + 180*a^10 - 210*a^8 + 180*a^4 - 120*a^2 + 24,\n",
       " a^60 - 12*a^50 - 30*a^44 + 70*a^42 + 6*a^40 + 360*a^38 - 390*a^36 - 1080*a^34 + 1740*a^32 - 720*a^30 - 105*a^28 + 330*a^26 + 580*a^24 - 1560*a^22 + 750*a^20 + 600*a^18 - 1140*a^16 + 840*a^14 - 930*a^12 + 1980*a^10 - 1530*a^8 - 600*a^6 + 1440*a^4 - 720*a^2 + 120,\n",
       " a^84 - 14*a^72 - 42*a^64 + 126*a^62 - 63*a^60 + 630*a^56 - 420*a^54 + 630*a^52 - 5124*a^50 + 1680*a^48 + 12600*a^46 - 15309*a^44 + 5488*a^42 + 42*a^40 + 2520*a^38 - 2905*a^36 - 7140*a^34 + 11550*a^32 - 3010*a^30 - 1470*a^28 - 3780*a^26 + 15470*a^24 - 25620*a^22 + 18270*a^20 + 4200*a^18 - 15540*a^16 + 11340*a^14 - 16170*a^12 + 23940*a^10 - 10080*a^8 - 10080*a^6 + 12600*a^4 - 5040*a^2 + 720,\n",
       " a^112 - 16*a^98 - 56*a^88 + 168*a^86 + 8*a^84 - 112*a^82 - 70*a^80 + 1008*a^78 - 1344*a^76 + 1680*a^74 + 1148*a^72 - 8400*a^70 + 1680*a^68 - 20160*a^66 + 64372*a^64 + 11032*a^62 - 151704*a^60 + 141120*a^58 - 35280*a^56 - 3696*a^54 + 5768*a^52 - 40992*a^50 + 12299*a^48 + 104776*a^46 - 126784*a^44 + 53088*a^42 - 28924*a^40 + 14280*a^38 + 98560*a^36 - 211680*a^34 + 105840*a^32 + 99680*a^30 - 80920*a^28 - 103600*a^26 + 239400*a^24 - 290640*a^22 + 178080*a^20 + 47040*a^18 - 127050*a^16 + 139440*a^14 - 270480*a^12 + 282240*a^10 - 45360*a^8 - 141120*a^6 + 120960*a^4 - 40320*a^2 + 5040,\n",
       " a^144 - 18*a^128 - 72*a^116 + 216*a^114 + 9*a^112 - 168*a^108 + 1260*a^104 - 2016*a^102 + 2880*a^98 + 4158*a^96 - 18144*a^94 + 22680*a^92 - 28560*a^90 - 45828*a^88 + 92160*a^86 - 20088*a^84 + 452592*a^82 - 794430*a^80 - 474768*a^78 + 1892520*a^76 - 1435392*a^74 + 373296*a^72 - 75600*a^70 + 13104*a^68 - 173376*a^66 + 568134*a^64 + 102816*a^62 - 1342656*a^60 + 1214640*a^58 - 275184*a^56 - 252672*a^54 + 491400*a^52 - 38808*a^50 - 1328922*a^48 + 1918224*a^46 - 1010520*a^44 + 620928*a^42 - 1473696*a^40 + 1658160*a^38 + 108920*a^36 - 1895040*a^34 + 525420*a^32 + 1975680*a^30 - 695520*a^28 - 3296160*a^26 + 5720400*a^24 - 4959360*a^22 + 1648080*a^20 + 1043280*a^18 - 1459080*a^16 + 2600640*a^14 - 4505760*a^12 + 3265920*a^10 + 272160*a^8 - 1935360*a^6 + 1270080*a^4 - 362880*a^2 + 40320,\n",
       " a^180 - 20*a^162 - 90*a^148 + 270*a^146 + 10*a^144 - 240*a^138 + 2160*a^134 - 3300*a^132 - 252*a^130 - 180*a^128 + 5040*a^126 + 3780*a^124 - 22680*a^122 + 25200*a^120 + 15120*a^118 - 51705*a^116 - 113850*a^114 + 302490*a^112 - 483840*a^110 + 359520*a^108 + 982800*a^106 - 743400*a^104 + 1037520*a^102 - 8957160*a^100 + 9327600*a^98 + 11079300*a^96 - 25583040*a^94 + 16553160*a^92 - 3901080*a^90 - 473040*a^88 + 915840*a^86 - 195420*a^84 + 4577580*a^82 - 8085294*a^80 - 4562880*a^78 + 18819360*a^76 - 14815080*a^74 + 4579680*a^72 - 1935360*a^70 + 5062680*a^68 - 7081620*a^66 - 2570400*a^64 + 21299040*a^62 - 28350420*a^60 + 20107080*a^58 - 8638560*a^56 - 12048960*a^54 + 33927390*a^52 - 15133860*a^50 - 35130270*a^48 + 46090800*a^46 - 15397200*a^44 + 6347040*a^42 - 18579960*a^40 + 11415600*a^38 + 10027500*a^36 - 10773000*a^34 - 21621600*a^32 + 44553600*a^30 - 9563400*a^28 - 62748000*a^26 + 107604000*a^24 - 83764800*a^22 + 21795480*a^20 + 11264400*a^18 - 22415400*a^16 + 55339200*a^14 - 74239200*a^12 + 36469440*a^10 + 14061600*a^8 - 27216000*a^6 + 14515200*a^4 - 3628800*a^2 + 362880]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# EGF of strongly connected labeled digraphs (we take a parameter a = 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": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0,\n",
       " 1,\n",
       " 1,\n",
       " 18,\n",
       " 1606,\n",
       " 565080,\n",
       " 734774776,\n",
       " 3523091615568,\n",
       " 63519209389664176,\n",
       " 4400410978376102609280,\n",
       " 1190433705317814685295399296]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Numbers of strongly connected labeled digraphs\n",
    "SCDsub = [scd[i][0][0].subs(a=sqrt(2)) * i.factorial() for i in srange(disp)]\n",
    "SCDsub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "disp = 5 # How many terms we want to display"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1,\n",
       " t,\n",
       " t^2 + (a^4 - 2*a^2 + 1)*t,\n",
       " t^3 + (3*a^4 - 6*a^2 + 3)*t^2 + (a^12 - 6*a^8 + 6*a^6 + 3*a^4 - 6*a^2 + 2)*t,\n",
       " t^4 + (6*a^4 - 12*a^2 + 6)*t^3 + (4*a^12 - 21*a^8 + 12*a^6 + 30*a^4 - 36*a^2 + 11)*t^2 + (a^24 - 8*a^18 - 6*a^16 + 36*a^14 - 20*a^12 - 21*a^8 + 12*a^6 + 24*a^4 - 24*a^2 + 6)*t]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# EGF of semi-strong labeled digraphs,\n",
    "# with the marking variable t for strongly connected components (we take a parameter a = sqrt{2})\n",
    "ssdt = exp(t*scd)\n",
    "SSDT = [ssdt[i] * i.factorial() for i in srange(disp)]\n",
    "SSDT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1,\n",
       " t,\n",
       " ((2*a^2 - 1)/a^2)*t^2 + ((a^4 - 2*a^2 + 1)/a^2)*t,\n",
       " ((6*a^6 - 6*a^4 + 1)/a^6)*t^3 + ((6*a^8 - 12*a^6 + 3*a^4 + 6*a^2 - 3)/a^6)*t^2 + ((a^12 - 6*a^8 + 6*a^6 + 3*a^4 - 6*a^2 + 2)/a^6)*t,\n",
       " ((24*a^12 - 36*a^10 + 6*a^8 + 8*a^6 - 1)/a^12)*t^4 + ((36*a^14 - 84*a^12 + 36*a^10 + 36*a^8 - 24*a^6 + 6*a^4 - 12*a^2 + 6)/a^12)*t^3 + ((8*a^18 + 6*a^16 - 72*a^14 + 80*a^12 - 21*a^8 + 4*a^6 - 30*a^4 + 36*a^2 - 11)/a^12)*t^2 + ((a^24 - 8*a^18 - 6*a^16 + 36*a^14 - 20*a^12 - 21*a^8 + 12*a^6 + 24*a^4 - 24*a^2 + 6)/a^12)*t]"
      ]
     },
     "execution_count": 14,
     "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{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": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1,\n",
       " t,\n",
       " 3/2*t^2 + 1/2*t,\n",
       " 25/8*t^3 + 21/8*t^2 + 9/4*t,\n",
       " 543/64*t^4 + 387/32*t^3 + 1173/64*t^2 + 803/32*t]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "DTsub = [sum(dt[i][0][k].subs(a=sqrt(2)) * i.factorial() * t^k for k in srange(N)) for i in srange(disp)]\n",
    "DTsub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, 2, 8, 64]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "DTsubsub = [DTsub[i].subs(t=1) for i in srange(disp)]\n",
    "DTsubsub"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"color:black; font-weight: bold; font-size:18px\">\n",
    "    Principal section\n",
    "</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1,\n",
       " s*t,\n",
       " (1/a^2*s^2 + ((2*a^2 - 2)/a^2)*s)*t^2 + (((a^4 - 2*a^2 + 1)/a^2)*s)*t,\n",
       " (1/a^6*s^3 + ((3*a^4 - 3)/a^6)*s^2 + ((6*a^6 - 9*a^4 + 3)/a^6)*s)*t^3 + (((3*a^4 - 6*a^2 + 3)/a^6)*s^2 + ((6*a^8 - 12*a^6 + 12*a^2 - 6)/a^6)*s)*t^2 + (((a^12 - 6*a^8 + 6*a^6 + 3*a^4 - 6*a^2 + 2)/a^6)*s)*t,\n",
       " (1/a^12*s^4 + ((4*a^6 - 4)/a^12)*s^3 + ((12*a^10 - 6*a^8 - 12*a^6 + 6)/a^12)*s^2 + ((24*a^12 - 48*a^10 + 12*a^8 + 16*a^6 - 4)/a^12)*s)*t^4 + (((6*a^4 - 12*a^2 + 6)/a^12)*s^3 + ((6*a^12 - 18*a^8 + 12*a^6 - 18*a^4 + 36*a^2 - 18)/a^12)*s^2 + ((36*a^14 - 90*a^12 + 36*a^10 + 54*a^8 - 36*a^6 + 18*a^4 - 36*a^2 + 18)/a^12)*s)*t^3 + (((4*a^12 - 21*a^8 + 12*a^6 + 30*a^4 - 36*a^2 + 11)/a^12)*s^2 + ((8*a^18 + 6*a^16 - 72*a^14 + 76*a^12 - 8*a^6 - 60*a^4 + 72*a^2 - 22)/a^12)*s)*t^2 + (((a^24 - 8*a^18 - 6*a^16 + 36*a^14 - 20*a^12 - 21*a^8 + 12*a^6 + 24*a^4 - 24*a^2 + 6)/a^12)*s)*t]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# EGF of labeled digraphs,\n",
    "# with the marking variable t for strongly connected components\n",
    "# and the marking variable s for source-like components (we take a parameter a = sqrt{2})\n",
    "dts = dt * exp_had_prod(exp((s-1)*t*scd),hatset,N)\n",
    "DTS = [dts[i] * i.factorial() for i in srange(disp)]\n",
    "DTS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1,\n",
       " s*t,\n",
       " 1/2*s^2*t^2 + s*t^2 + 1/2*s*t,\n",
       " 1/8*s^3*t^3 + 9/8*s^2*t^3 + 3/8*s^2*t^2 + 15/8*s*t^3 + 9/4*s*t^2 + 9/4*s*t,\n",
       " 1/64*s^4*t^4 + 7/16*s^3*t^4 + 3/32*s^3*t^3 + 99/32*s^2*t^4 + 87/32*s^2*t^3 + 79/16*s*t^4 + 75/64*s^2*t^2 + 297/32*s*t^3 + 549/32*s*t^2 + 803/32*s*t]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "DTSsub = [sum(sum(dts[i][0][k][l].subs(a=sqrt(2)) * i.factorial() * s^l * t^k for l in srange(N)) for k in srange(N)) for i in srange(disp)]\n",
    "DTSsub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1,\n",
       " t,\n",
       " 3/2*t^2 + 1/2*t,\n",
       " 25/8*t^3 + 21/8*t^2 + 9/4*t,\n",
       " 543/64*t^4 + 387/32*t^3 + 1173/64*t^2 + 803/32*t]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "DTSsubsub = [DTSsub[i].subs(s=1) for i in srange(disp)]\n",
    "DTSsubsub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[t,\n",
       " -2*a^2*t*w,\n",
       " ((-2*a^6 + 6*a^4)*t)*w^2 + (-2*a^2*t^2 + 2*a^2*t)*w,\n",
       " ((-2*a^12 + 18*a^8 - 24*a^6)*t)*w^3 + (12*a^2*t^2 - 12*a^2*t)*w^2,\n",
       " ((-2*a^20 + 24*a^14 + 18*a^12 - 144*a^10 + 120*a^8)*t)*w^4 + ((24*a^4 - 72*a^2)*t^2 + (-24*a^4 + 72*a^2)*t)*w^3 + (12*t^3 + (-12*a^4 + 24*a^2 - 36)*t^2 + (12*a^4 - 24*a^2 + 24)*t)*w^2]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Preparation for calculation of CGF of labeled digraphs,\n",
    "# with the marking variable t for strongly connected components (we take a parameter a = sqrt{2})\n",
    "num1 = exp_had_prod(PhiTwoTwoOne,t*exp((1-t)*scd.subs(z=a^3*z^2*w))*exp_had_prod(PhiTwoOneTwo,(1/g.subs(z=a^2*z*w))*(1/g.subs(z=a^2*z*w)),N),N)\n",
    "NUM1 = [num1[i] * i.factorial() for i in srange(disp)]\n",
    "NUM1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[t,\n",
       " (2*a^2*t^2 - 2*a^2*t)*w,\n",
       " ((6*a^4 - 2*a^2)*t^3 + (2*a^6 - 12*a^4 + 2*a^2)*t^2 + (-2*a^6 + 6*a^4)*t)*w^2 + (-2*a^2*t^2 + 2*a^2*t)*w,\n",
       " ((24*a^6 - 18*a^4 + 2)*t^4 + (18*a^8 - 72*a^6 + 24*a^4 + 12*a^2 - 6)*t^3 + (2*a^12 - 36*a^8 + 72*a^6 - 6*a^4 - 12*a^2 + 4)*t^2 + (-2*a^12 + 18*a^8 - 24*a^6)*t)*w^3 + (-12*a^4*t^3 + (12*a^4 + 12*a^2)*t^2 - 12*a^2*t)*w^2,\n",
       " (((120*a^12 - 144*a^10 + 18*a^8 + 24*a^6 - 2)/a^4)*t^5 + ((144*a^14 - 516*a^12 + 288*a^10 + 108*a^8 - 88*a^6 + 12*a^4 - 24*a^2 + 12)/a^4)*t^4 + ((24*a^18 + 18*a^16 - 432*a^14 + 772*a^12 - 168*a^10 - 180*a^8 + 72*a^6 - 60*a^4 + 72*a^2 - 22)/a^4)*t^3 + ((2*a^24 - 48*a^18 - 36*a^16 + 432*a^14 - 496*a^12 + 24*a^10 + 54*a^8 - 8*a^6 + 48*a^4 - 48*a^2 + 12)/a^4)*t^2 + (-2*a^20 + 24*a^14 + 18*a^12 - 144*a^10 + 120*a^8)*t)*w^4 + ((-72*a^6 + 24*a^4)*t^4 + (-24*a^8 + 120*a^6 + 48*a^4)*t^3 + (24*a^8 - 48*a^6 - 48*a^4 - 72*a^2)*t^2 + (-24*a^4 + 72*a^2)*t)*w^3 + (12*t^3 + (-12*a^4 + 24*a^2 - 36)*t^2 + (12*a^4 - 24*a^2 + 24)*t)*w^2]"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# CGF of labeled digraphs,\n",
    "# with the marking variable t for strongly connected components (we take a parameter a = sqrt{2})\n",
    "qdt = dt.subs(z=a^2*z*w)*dt.subs(z=a^2*z*w)*num1\n",
    "QDT = [qdt[i] * i.factorial() for i in srange(disp)]\n",
    "QDT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(s - 1)*t,\n",
       " ((-2*a^2*s + 2*a^2)*t)*w,\n",
       " (((-2*a^6 + 6*a^4)*s + 2*a^6 - 6*a^4)*t)*w^2 + ((2*a^2*s^2 - 4*a^2*s + 2*a^2)*t^2 + (2*a^2*s - 2*a^2)*t)*w,\n",
       " (((-2*a^12 + 18*a^8 - 24*a^6)*s + 2*a^12 - 18*a^8 + 24*a^6)*t)*w^3 + ((-12*a^2*s^2 + 24*a^2*s - 12*a^2)*t^2 + (-12*a^2*s + 12*a^2)*t)*w^2,\n",
       " (((-2*a^20 + 24*a^14 + 18*a^12 - 144*a^10 + 120*a^8)*s + 2*a^20 - 24*a^14 - 18*a^12 + 144*a^10 - 120*a^8)*t)*w^4 + (((-24*a^4 + 72*a^2)*s^2 + (48*a^4 - 144*a^2)*s - 24*a^4 + 72*a^2)*t^2 + ((-24*a^4 + 72*a^2)*s + 24*a^4 - 72*a^2)*t)*w^3 + ((12*s^3 - 36*s^2 + 36*s - 12)*t^3 + ((12*a^4 - 24*a^2 + 36)*s^2 + (-24*a^4 + 48*a^2 - 72)*s + 12*a^4 - 24*a^2 + 36)*t^2 + ((12*a^4 - 24*a^2 + 24)*s - 12*a^4 + 24*a^2 - 24)*t)*w^2]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Preparation for calculation of CGF of labeled digraphs,\n",
    "# with the marking variable t for strongly connected components\n",
    "# and the marking variable s for source-like components (we take a parameter a = sqrt{2})\n",
    "num2 = exp_had_prod(PhiTwoTwoOne,(s-1)*t*exp((1-t+t*s)*scd.subs(z=a^3*z^2*w))*exp_had_prod(PhiTwoOneTwo,(1/g.subs(z=a^2*z*w))*(1/g.subs(z=a^2*z*w)),N),N)\n",
    "NUM2 = [num2[i] * i.factorial() for i in srange(disp)]\n",
    "NUM2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[s*t,\n",
       " (2*a^2*s*t^2 - 2*a^2*s*t)*w,\n",
       " ((a^2*s^2 + (6*a^4 - 3*a^2)*s)*t^3 + ((2*a^6 - 12*a^4 + 2*a^2)*s)*t^2 + ((-2*a^6 + 6*a^4)*s)*t)*w^2 + ((2*a^2*s^2 - 4*a^2*s)*t^2 + 2*a^2*s*t)*w,\n",
       " ((s^3 + (6*a^4 - 3)*s^2 + (24*a^6 - 24*a^4 + 4)*s)*t^4 + ((-3*a^4 - 6*a^2 + 3)*s^2 + (18*a^8 - 72*a^6 + 27*a^4 + 18*a^2 - 9)*s)*t^3 + ((2*a^12 - 36*a^8 + 72*a^6 - 6*a^4 - 12*a^2 + 4)*s)*t^2 + ((-2*a^12 + 18*a^8 - 24*a^6)*s)*t)*w^3 + ((6*a^4*s^2 - 18*a^4*s)*t^3 + (-12*a^2*s^2 + (12*a^4 + 24*a^2)*s)*t^2 - 12*a^2*s*t)*w^2,\n",
       " ((1/a^4*s^4 + ((8*a^6 - 4)/a^4)*s^3 + ((36*a^10 - 12*a^8 - 24*a^6 + 6)/a^4)*s^2 + ((120*a^12 - 180*a^10 + 30*a^8 + 40*a^6 - 5)/a^4)*s)*t^5 + (((-8*a^6 + 6*a^4 - 12*a^2 + 6)/a^4)*s^3 + ((12*a^12 - 48*a^10 - 36*a^8 + 48*a^6 - 18*a^4 + 36*a^2 - 18)/a^4)*s^2 + ((144*a^14 - 528*a^12 + 336*a^10 + 144*a^8 - 128*a^6 + 24*a^4 - 48*a^2 + 24)/a^4)*s)*t^4 + (((-8*a^12 + 12*a^10 + 27*a^8 - 12*a^6 + 30*a^4 - 36*a^2 + 11)/a^4)*s^2 + ((24*a^18 + 18*a^16 - 432*a^14 + 780*a^12 - 180*a^10 - 207*a^8 + 84*a^6 - 90*a^4 + 108*a^2 - 33)/a^4)*s)*t^3 + (((2*a^24 - 48*a^18 - 36*a^16 + 432*a^14 - 496*a^12 + 24*a^10 + 54*a^8 - 8*a^6 + 48*a^4 - 48*a^2 + 12)/a^4)*s)*t^2 + ((-2*a^20 + 24*a^14 + 18*a^12 - 144*a^10 + 120*a^8)*s)*t)*w^4 + (((24*a^6 - 24*a^4)*s^2 + (-96*a^6 + 48*a^4)*s)*t^4 + ((12*a^8 - 24*a^6 - 24*a^4)*s^2 + (-36*a^8 + 144*a^6 + 72*a^4)*s)*t^3 + ((-24*a^4 + 72*a^2)*s^2 + (24*a^8 - 48*a^6 - 24*a^4 - 144*a^2)*s)*t^2 + ((-24*a^4 + 72*a^2)*s)*t)*w^3 + ((12*s^3 - 36*s^2 + 36*s)*t^3 + ((12*a^4 - 24*a^2 + 36)*s^2 + (-24*a^4 + 48*a^2 - 72)*s)*t^2 + ((12*a^4 - 24*a^2 + 24)*s)*t)*w^2]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# CGF of labeled digraphs,\n",
    "# with the marking variable t for strongly connected components\n",
    "# and the marking variable s for source-like components (we take a parameter a = sqrt{2})\n",
    "qdts = dt.subs(z=a^2*z*w)*(num2 + dts.subs(z=a^2*z*w)*num1)\n",
    "QDTS = [qdts[i] * i.factorial() for i in srange(disp)]\n",
    "QDTS\n",
    "# For N = 12, this calculation took about 2 minutes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 + a^2*s*t*w*z + (((1/2*a^2*s^2 + (a^4 - a^2)*s)*t^2 + ((1/2*a^6 - a^4 + 1/2*a^2)*s)*t)*w^2)*z^2 + (((1/6*s^3 + (1/2*a^4 - 1/2)*s^2 + (a^6 - 3/2*a^4 + 1/2)*s)*t^3 + ((1/2*a^4 - a^2 + 1/2)*s^2 + (a^8 - 2*a^6 + 2*a^2 - 1)*s)*t^2 + ((1/6*a^12 - a^8 + a^6 + 1/2*a^4 - a^2 + 1/3)*s)*t)*w^3)*z^3 + (((1/24/a^4*s^4 + ((1/6*a^6 - 1/6)/a^4)*s^3 + ((1/2*a^10 - 1/4*a^8 - 1/2*a^6 + 1/4)/a^4)*s^2 + ((a^12 - 2*a^10 + 1/2*a^8 + 2/3*a^6 - 1/6)/a^4)*s)*t^4 + (((1/4*a^4 - 1/2*a^2 + 1/4)/a^4)*s^3 + ((1/4*a^12 - 3/4*a^8 + 1/2*a^6 - 3/4*a^4 + 3/2*a^2 - 3/4)/a^4)*s^2 + ((3/2*a^14 - 15/4*a^12 + 3/2*a^10 + 9/4*a^8 - 3/2*a^6 + 3/4*a^4 - 3/2*a^2 + 3/4)/a^4)*s)*t^3 + (((1/6*a^12 - 7/8*a^8 + 1/2*a^6 + 5/4*a^4 - 3/2*a^2 + 11/24)/a^4)*s^2 + ((1/3*a^18 + 1/4*a^16 - 3*a^14 + 19/6*a^12 - 1/3*a^6 - 5/2*a^4 + 3*a^2 - 11/12)/a^4)*s)*t^2 + (((1/24*a^24 - 1/3*a^18 - 1/4*a^16 + 3/2*a^14 - 5/6*a^12 - 7/8*a^8 + 1/2*a^6 + a^4 - a^2 + 1/4)/a^4)*s)*t)*w^4)*z^4 + (((1/120/a^10*s^5 + ((1/24*a^8 - 1/24)/a^10)*s^4 + ((1/6*a^14 - 1/12*a^12 - 1/6*a^8 + 1/12)/a^10)*s^3 + ((1/2*a^18 - 1/2*a^16 - 1/2*a^14 + 1/3*a^12 + 1/4*a^8 - 1/12)/a^10)*s^2 + ((a^20 - 5/2*a^18 + 5/4*a^16 + 5/6*a^14 - 5/12*a^12 - 5/24*a^8 + 1/24)/a^10)*s)*t^5 + (((1/12*a^4 - 1/6*a^2 + 1/12)/a^10)*s^4 + ((1/12*a^16 - 1/6*a^14 + 1/3*a^12 - 1/2*a^10 + 1/4*a^8 - 1/3*a^4 + 2/3*a^2 - 1/3)/a^10)*s^3 + ((1/2*a^20 - 1/2*a^18 - 5/4*a^16 + 2*a^14 - 3/2*a^12 + 3/2*a^10 - 3/4*a^8 + 1/2*a^4 - a^2 + 1/2)/a^10)*s^2 + ((2*a^22 - 6*a^20 + 4*a^18 + 10/3*a^16 - 14/3*a^14 + 7/3*a^12 - 2*a^10 + a^8 - 1/3*a^4 + 2/3*a^2 - 1/3)/a^10)*s)*t^4 + (((1/12*a^12 - 3/8*a^8 + a^4 - a^2 + 7/24)/a^10)*s^3 + ((1/12*a^24 - 1/12*a^20 - 1/2*a^18 + 7/8*a^16 - a^14 + 17/12*a^12 - 3/2*a^10 + 19/12*a^8 - 3*a^4 + 3*a^2 - 7/8)/a^10)*s^2 + ((1/2*a^26 + 1/2*a^24 - 6*a^22 + 31/4*a^20 - 39/8*a^16 + 4*a^14 - 19/4*a^12 + 9/2*a^10 - 5/2*a^8 + 3*a^4 - 3*a^2 + 7/8)/a^10)*s)*t^3 + (((1/24*a^24 - 1/3*a^18 - 1/6*a^16 + 4/3*a^14 - 5/4*a^12 + 3/2*a^10 - 17/8*a^8 + 29/12*a^4 - 11/6*a^2 + 5/12)/a^10)*s^2 + ((1/12*a^32 + 1/6*a^28 - a^26 - 17/12*a^24 + 6*a^22 - 25/6*a^20 - 1/3*a^18 + 17/12*a^16 - 10/3*a^14 + 29/6*a^12 - 5*a^10 + 19/4*a^8 - 29/6*a^4 + 11/3*a^2 - 5/6)/a^10)*s)*t^2 + (((1/120*a^40 - 1/12*a^32 - 1/6*a^28 + 1/2*a^26 + 19/24*a^24 - 2*a^22 + a^20 - 1/3*a^18 - 1/6*a^16 + 4/3*a^14 - 4/3*a^12 + 3/2*a^10 - 7/4*a^8 + 3/2*a^4 - a^2 + 1/5)/a^10)*s)*t)*w^5)*z^5 + (((1/720/a^18*s^6 + ((1/120*a^10 - 1/120)/a^18)*s^5 + ((1/24*a^18 - 1/48*a^16 - 1/24*a^10 + 1/48)/a^18)*s^4 + ((1/6*a^24 - 1/6*a^22 - 5/36*a^18 + 1/12*a^16 + 1/12*a^10 - 1/36)/a^18)*s^3 + ((1/2*a^28 - 3/4*a^26 - 3/8*a^24 + 2/3*a^22 + 1/6*a^18 - 7/48*a^16 - 1/12*a^10 + 1/48)/a^18)*s^2 + ((a^30 - 3*a^28 + 9/4*a^26 + 3/4*a^24 - a^22 - 1/6*a^18 + 1/8*a^16 + 1/20*a^10 - 1/120)/a^18)*s)*t^6 + (((1/48*a^4 - 1/24*a^2 + 1/48)/a^18)*s^5 + ((1/48*a^20 - 1/24*a^18 + 1/48*a^16 + 1/12*a^14 - 1/6*a^12 + 1/12*a^10 - 5/48*a^4 + 5/24*a^2 - 5/48)/a^18)*s^4 + ((1/6*a^26 - 1/3*a^24 + 1/3*a^22 - 13/24*a^20 + 7/12*a^18 - 5/24*a^16 - 1/3*a^14 + 2/3*a^12 - 1/3*a^10 + 5/24*a^4 - 5/12*a^2 + 5/24)/a^18)*s^3 + ((3/4*a^30 - 5/4*a^28 - 5/4*a^26 + 13/4*a^24 - 23/12*a^22 + 35/24*a^20 - 5/3*a^18 + 5/8*a^16 + 1/2*a^14 - a^12 + 1/2*a^10 - 5/24*a^4 + 5/12*a^2 - 5/24)/a^18)*s^2 + ((5/2*a^32 - 35/4*a^30 + 65/8*a^28 + 10/3*a^26 - 205/24*a^24 + 25/6*a^22 - 115/48*a^20 + 55/24*a^18 - 35/48*a^16 - 5/12*a^14 + 5/6*a^12 - 5/12*a^10 + 5/48*a^4 - 5/24*a^2 + 5/48)/a^18)*s)*t^5 + (((1/36*a^12 - 5/48*a^8 - 1/12*a^6 + 11/24*a^4 - 5/12*a^2 + 17/144)/a^18)*s^4 + ((1/36*a^30 - 1/6*a^26 + 7/24*a^24 - 1/3*a^22 + 7/12*a^20 - 59/72*a^18 + 1/8*a^16 + a^14 - 10/9*a^12 + 7/24*a^10 + 5/12*a^8 + 1/3*a^6 - 11/6*a^4 + 5/3*a^2 - 17/36)/a^18)*s^3 + ((1/6*a^34 + 1/8*a^32 - 11/12*a^30 - 5/12*a^28 + 19/8*a^26 - 11/8*a^24 + 13/12*a^22 - 7/2*a^20 + 25/6*a^18 - 5/6*a^16 - 3*a^14 + 19/6*a^12 - 7/8*a^10 - 5/8*a^8 - 1/2*a^6 + 11/4*a^4 - 5/2*a^2 + 17/24)/a^18)*s^2 + ((2/3*a^36 + 5/6*a^34 - 21/2*a^32 + 148/9*a^30 - 8/3*a^28 - 67/6*a^26 + 33/4*a^24 - 17/3*a^22 + 53/6*a^20 - 137/18*a^18 + 17/12*a^16 + 4*a^14 - 37/9*a^12 + 7/6*a^10 + 5/12*a^8 + 1/3*a^6 - 11/6*a^4 + 5/3*a^2 - 17/36)/a^18)*s)*t^4 + (((1/48*a^24 - 1/6*a^18 - 1/24*a^16 + 7/12*a^14 - 13/16*a^12 + 11/8*a^10 - 11/8*a^8 - 2/3*a^6 + 107/48*a^4 - 35/24*a^2 + 5/16)/a^18)*s^3 + ((1/48*a^40 - 1/24*a^34 - 5/24*a^32 + 1/6*a^30 + 19/48*a^28 - 7/24*a^26 + 1/48*a^24 - 4/3*a^22 + 169/48*a^20 - 19/6*a^18 + 23/48*a^16 + 2/3*a^14 + 29/48*a^12 - 89/24*a^10 + 33/8*a^8 + 2*a^6 - 107/16*a^4 + 35/8*a^2 - 15/16)/a^18)*s^2 + ((1/8*a^42 - 1/16*a^40 + 1/2*a^38 - 15/8*a^36 - 31/8*a^34 + 16*a^32 - 13*a^30 - 37/16*a^28 + 6*a^26 - 57/16*a^24 + 8*a^22 - 217/16*a^20 + 45/4*a^18 - 19/16*a^16 - 11/2*a^14 + 49/16*a^12 + 23/8*a^10 - 33/8*a^8 - 2*a^6 + 107/16*a^4 - 35/8*a^2 + 15/16)/a^18)*s)*t^3 + (((1/120*a^40 - 1/12*a^32 - 7/48*a^28 + 11/24*a^26 + 119/144*a^24 - 13/6*a^22 + 25/24*a^20 + 2/3*a^18 - 13/8*a^16 + 7/4*a^14 - 307/144*a^12 + 33/8*a^10 - 163/48*a^8 - 17/12*a^6 + 91/24*a^4 - 25/12*a^2 + 137/360)/a^18)*s^2 + ((1/60*a^50 + 1/24*a^44 - 2/9*a^42 + 1/40*a^40 - a^38 + 7/4*a^36 + 53/12*a^34 - 41/4*a^32 + 95/18*a^30 + 11/12*a^28 - 17/12*a^26 - 7/36*a^24 + 1/2*a^22 + 7/2*a^20 - 56/9*a^18 + 7/2*a^16 - 1/2*a^14 + 163/72*a^12 - 157/20*a^10 + 163/24*a^8 + 17/6*a^6 - 91/12*a^4 + 25/6*a^2 - 137/180)/a^18)*s)*t^2 + (((1/720*a^60 - 1/60*a^50 - 1/24*a^44 + 7/72*a^42 + 1/120*a^40 + 1/2*a^38 - 13/24*a^36 - 3/2*a^34 + 29/12*a^32 - a^30 - 7/48*a^28 + 11/24*a^26 + 29/36*a^24 - 13/6*a^22 + 25/24*a^20 + 5/6*a^18 - 19/12*a^16 + 7/6*a^14 - 31/24*a^12 + 11/4*a^10 - 17/8*a^8 - 5/6*a^6 + 2*a^4 - a^2 + 1/6)/a^18)*s)*t)*w^6)*z^6 + (((1/5040/a^28*s^7 + ((1/720*a^12 - 1/720)/a^28)*s^6 + ((1/120*a^22 - 1/240*a^20 - 1/120*a^12 + 1/240)/a^28)*s^5 + ((1/24*a^30 - 1/24*a^28 + 1/144*a^24 - 1/24*a^22 + 1/48*a^20 + 1/48*a^12 - 1/144)/a^28)*s^4 + ((1/6*a^36 - 1/4*a^34 + 1/24*a^32 - 1/9*a^30 + 1/6*a^28 - 5/144*a^24 + 1/12*a^22 - 1/24*a^20 - 1/36*a^12 + 1/144)/a^28)*s^3 + ((1/2*a^40 - a^38 - 1/8*a^36 + a^34 - 5/24*a^32 + 1/12*a^30 - 7/24*a^28 + 1/16*a^24 - 1/12*a^22 + 11/240*a^20 + 1/48*a^12 - 1/240)/a^28)*s^2 + ((a^42 - 7/2*a^40 + 7/2*a^38 + 7/24*a^36 - 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853/20*a^8 + 1473/28*a^6 - 61921/2520*a^4 + 7129/1260*a^2 - 671/1260)/a^88)*s)*t^2 + (((1/39916800*a^220 - 1/1814400*a^200 - 1/362880*a^184 + 1/120960*a^182 + 1/3628800*a^180 - 1/120960*a^172 + 1/13440*a^168 - 1/10080*a^166 - 1/60480*a^164 - 1/181440*a^162 - 1/43200*a^160 + 1/5040*a^158 + 1/6720*a^156 - 1/840*a^154 + 3/2240*a^152 + 1/4800*a^150 + 487/725760*a^148 - 491/181440*a^146 - 1133/362880*a^144 + 1/96*a^142 - 43/5760*a^140 - 253/15120*a^138 + 19/1440*a^136 + 671/10080*a^134 - 17753/120960*a^132 + 38917/151200*a^130 - 3503/60480*a^128 - 299/720*a^126 + 22933/241920*a^124 - 161/160*a^122 + 33823/8064*a^120 - 4751/1920*a^118 - 111179/20160*a^116 + 60269/6720*a^114 - 118949/24192*a^112 + 437/504*a^110 + 271/2688*a^108 + 2671/10080*a^106 - 671/3360*a^104 + 313/1080*a^102 - 747179/302400*a^100 + 3407/1344*a^98 + 15803/5040*a^96 - 15461/2160*a^94 + 1759/384*a^92 - 41017/50400*a^90 - 2531/6048*a^88 + 2291/2520*a^86 - 23483/10080*a^84 + 23233/8640*a^82 + 24601/14400*a^80 - 35891/4320*a^78 + 41977/4320*a^76 - 20689/4320*a^74 - 311/270*a^72 + 1093/864*a^70 + 1883/288*a^68 - 8633/864*a^66 - 1907/1728*a^64 + 191/15*a^62 - 36235/3456*a^60 + 1073/540*a^58 + 5309/960*a^56 - 129613/8640*a^54 + 328531/17280*a^52 - 2527/576*a^50 - 71707/4320*a^48 + 13217/720*a^46 - 27581/2880*a^44 + 977/72*a^42 - 77699/4320*a^40 + 5135/864*a^38 + 3659/864*a^36 + 287/48*a^34 - 133/6*a^32 + 2813/144*a^30 + 1903/288*a^28 - 707/18*a^26 + 2491/48*a^24 - 1601/48*a^22 + 767/96*a^20 + 39/16*a^18 - 617/48*a^16 + 361/12*a^14 - 731/24*a^12 + 37/4*a^10 + 17/2*a^8 - 10*a^6 + 9/2*a^4 - a^2 + 1/11)/a^88)*s)*t)*w^11)*z^11 + O(z^12)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dtsqsub = dts.subs(z=a^2*z*w)\n",
    "dtsqsub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 + a^2*t*w*z + (((a^4 - 1/2*a^2)*t^2 + (1/2*a^6 - a^4 + 1/2*a^2)*t)*w^2)*z^2 + (((a^6 - a^4 + 1/6)*t^3 + (a^8 - 2*a^6 + 1/2*a^4 + a^2 - 1/2)*t^2 + (1/6*a^12 - a^8 + a^6 + 1/2*a^4 - a^2 + 1/3)*t)*w^3)*z^3 + ((((a^12 - 3/2*a^10 + 1/4*a^8 + 1/3*a^6 - 1/24)/a^4)*t^4 + ((3/2*a^14 - 7/2*a^12 + 3/2*a^10 + 3/2*a^8 - a^6 + 1/4*a^4 - 1/2*a^2 + 1/4)/a^4)*t^3 + ((1/3*a^18 + 1/4*a^16 - 3*a^14 + 10/3*a^12 - 7/8*a^8 + 1/6*a^6 - 5/4*a^4 + 3/2*a^2 - 11/24)/a^4)*t^2 + ((1/24*a^24 - 1/3*a^18 - 1/4*a^16 + 3/2*a^14 - 5/6*a^12 - 7/8*a^8 + 1/2*a^6 + a^4 - a^2 + 1/4)/a^4)*t)*w^4)*z^4 + ((((a^20 - 2*a^18 + 3/4*a^16 + 1/2*a^14 - 1/6*a^12 - 1/12*a^8 + 1/120)/a^10)*t^5 + ((2*a^22 - 11/2*a^20 + 7/2*a^18 + 13/6*a^16 - 17/6*a^14 + 7/6*a^12 - a^10 + 1/2*a^8 - 1/12*a^4 + 1/6*a^2 - 1/12)/a^10)*t^4 + ((1/2*a^26 + 7/12*a^24 - 6*a^22 + 23/3*a^20 - 1/2*a^18 - 4*a^16 + 3*a^14 - 13/4*a^12 + 3*a^10 - 31/24*a^8 + a^4 - a^2 + 7/24)/a^10)*t^3 + ((1/12*a^32 + 1/6*a^28 - a^26 - 11/8*a^24 + 6*a^22 - 25/6*a^20 - 2/3*a^18 + 5/4*a^16 - 2*a^14 + 43/12*a^12 - 7/2*a^10 + 21/8*a^8 - 29/12*a^4 + 11/6*a^2 - 5/12)/a^10)*t^2 + ((1/120*a^40 - 1/12*a^32 - 1/6*a^28 + 1/2*a^26 + 19/24*a^24 - 2*a^22 + a^20 - 1/3*a^18 - 1/6*a^16 + 4/3*a^14 - 4/3*a^12 + 3/2*a^10 - 7/4*a^8 + 3/2*a^4 - a^2 + 1/5)/a^10)*t)*w^5)*z^5 + ((((a^30 - 5/2*a^28 + 3/2*a^26 + 13/24*a^24 - 1/2*a^22 - 7/72*a^18 + 1/24*a^16 + 1/60*a^10 - 1/720)/a^18)*t^6 + ((5/2*a^32 - 8*a^30 + 55/8*a^28 + 9/4*a^26 - 45/8*a^24 + 31/12*a^22 - 35/24*a^20 + 7/6*a^18 - 7/24*a^16 - 1/6*a^14 + 1/3*a^12 - 1/6*a^10 + 1/48*a^4 - 1/24*a^2 + 1/48)/a^18)*t^5 + ((2/3*a^36 + a^34 - 83/8*a^32 + 140/9*a^30 - 37/12*a^28 - 215/24*a^26 + 43/6*a^24 - 59/12*a^22 + 71/12*a^20 - 307/72*a^18 + 17/24*a^16 + 2*a^14 - 73/36*a^12 + 7/12*a^10 + 5/48*a^8 + 1/12*a^6 - 11/24*a^4 + 5/12*a^2 - 17/144)/a^18)*t^4 + ((1/8*a^42 - 1/24*a^40 + 1/2*a^38 - 15/8*a^36 - 47/12*a^34 + 379/24*a^32 - 77/6*a^30 - 23/12*a^28 + 137/24*a^26 - 169/48*a^24 + 20/3*a^22 - 241/24*a^20 + 95/12*a^18 - 3/4*a^16 - 17/4*a^14 + 137/48*a^12 + 13/24*a^10 - 11/8*a^8 - 2/3*a^6 + 107/48*a^4 - 35/24*a^2 + 5/16)/a^18)*t^3 + ((1/60*a^50 + 1/24*a^44 - 2/9*a^42 + 1/30*a^40 - a^38 + 7/4*a^36 + 53/12*a^34 - 31/3*a^32 + 95/18*a^30 + 37/48*a^28 - 23/24*a^26 + 91/144*a^24 - 5/3*a^22 + 109/24*a^20 - 50/9*a^18 + 15/8*a^16 + 5/4*a^14 + 19/144*a^12 - 149/40*a^10 + 163/48*a^8 + 17/12*a^6 - 91/24*a^4 + 25/12*a^2 - 137/360)/a^18)*t^2 + ((1/720*a^60 - 1/60*a^50 - 1/24*a^44 + 7/72*a^42 + 1/120*a^40 + 1/2*a^38 - 13/24*a^36 - 3/2*a^34 + 29/12*a^32 - a^30 - 7/48*a^28 + 11/24*a^26 + 29/36*a^24 - 13/6*a^22 + 25/24*a^20 + 5/6*a^18 - 19/12*a^16 + 7/6*a^14 - 31/24*a^12 + 11/4*a^10 - 17/8*a^8 - 5/6*a^6 + 2*a^4 - a^2 + 1/6)/a^18)*t)*w^6)*z^6 + ((((a^42 - 3*a^40 + 5/2*a^38 + 1/3*a^36 - a^34 + 1/8*a^32 - 1/12*a^30 + 1/8*a^28 - 1/72*a^24 + 1/40*a^22 - 1/120*a^20 - 1/360*a^12 + 1/5040)/a^28)*t^7 + ((3*a^44 - 11*a^42 + 12*a^40 + a^38 - 77/8*a^36 + 23/4*a^34 - 5/2*a^32 + 9/4*a^30 - 3/4*a^28 - 1/2*a^26 + 43/60*a^24 - 13/30*a^22 + 11/120*a^20 + 1/24*a^16 - 1/12*a^14 + 1/24*a^12 - 1/240*a^4 + 1/120*a^2 - 1/240)/a^28)*t^6 + ((5/6*a^48 + 3/2*a^46 - 131/8*a^44 + 57/2*a^42 - 79/8*a^40 - 17*a^38 + 1285/72*a^36 - 21/2*a^34 + 137/12*a^32 - 69/8*a^30 + 11/24*a^28 + 59/12*a^26 - 359/72*a^24 + 53/24*a^22 - 1/6*a^20 + 1/6*a^18 - 11/12*a^16 + 5/6*a^14 - 11/48*a^12 - 1/48*a^8 - 1/24*a^6 + 7/48*a^4 - 1/8*a^2 + 5/144)/a^28)*t^5 + ((1/6*a^54 - 1/8*a^52 + a^50 - 221/72*a^48 - 65/8*a^46 + 101/3*a^44 - 1141/36*a^42 - 43/12*a^40 + 245/12*a^38 - 1103/72*a^36 + 239/12*a^34 - 605/24*a^32 + 353/24*a^30 + 11/4*a^28 - 38/3*a^26 + 1481/144*a^24 - 23/12*a^22 - 49/24*a^20 - 23/18*a^18 + 107/24*a^16 - 37/12*a^14 + 137/144*a^12 - 5/8*a^10 + 11/24*a^8 + 7/12*a^6 - 19/16*a^4 + 17/24*a^2 - 7/48)/a^28)*t^4 + ((1/40*a^62 - 1/120*a^60 + 1/8*a^56 - 5/12*a^54 + 11/40*a^52 - 35/12*a^50 + 34/9*a^48 + 337/24*a^46 - 785/24*a^44 + 73/4*a^42 + 1241/240*a^40 - 35/4*a^38 + 35/3*a^36 - 259/12*a^34 + 557/24*a^32 - 51/4*a^30 - 103/48*a^28 + 271/24*a^26 - 247/36*a^24 - 103/20*a^22 + 539/80*a^20 + 43/12*a^18 - 73/8*a^16 + 139/24*a^14 - 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1/24*a^44 + 5/24*a^42 - 1/32*a^40 - 1/24*a^38 + 1/30*a^36 - 1/40*a^34 + 1/576*a^32 + 1/360*a^30 - 1/240*a^26 + 1/720*a^24 + 1/2520*a^14 - 1/40320)/a^40)*t^8 + ((7/2*a^58 - 29/2*a^56 + 77/4*a^54 - 31/12*a^52 - 175/12*a^50 + 35/3*a^48 - 107/24*a^46 + 41/12*a^44 - 19/12*a^42 - 5/6*a^40 + 79/60*a^38 - 217/240*a^36 + 101/360*a^34 + 37/720*a^32 + 19/720*a^30 - 53/360*a^28 + 77/720*a^26 - 1/45*a^24 - 1/120*a^18 + 1/60*a^16 - 1/120*a^14 + 1/1440*a^4 - 1/720*a^2 + 1/1440)/a^40)*t^7 + ((a^62 + 25/12*a^60 - 97/4*a^58 + 387/8*a^56 - 295/12*a^54 - 109/4*a^52 + 116/3*a^50 - 173/8*a^48 + 143/8*a^46 - 1061/72*a^44 + 223/180*a^42 + 863/96*a^40 - 1217/120*a^38 + 3857/720*a^36 - 83/240*a^34 - 599/1440*a^32 - 61/45*a^30 + 11/6*a^28 - 121/144*a^26 + 5/36*a^24 - 1/24*a^22 - 1/12*a^20 + 7/24*a^18 - 1/4*a^16 + 5/72*a^14 - 1/720*a^12 + 1/320*a^8 + 1/80*a^6 - 17/480*a^4 + 7/240*a^2 - 23/2880)/a^40)*t^6 + ((5/24*a^68 - 1/4*a^66 + 163/96*a^64 - 113/24*a^62 - 29/2*a^60 + 1535/24*a^58 - 20119/288*a^56 - 97/72*a^54 + 417/8*a^52 - 6155/144*a^50 + 11399/288*a^48 - 425/9*a^46 + 4067/144*a^44 + 427/72*a^42 - 8051/288*a^40 + 413/16*a^38 - 889/144*a^36 - 61/9*a^34 + 19/16*a^32 + 1193/144*a^30 - 601/72*a^28 + 533/144*a^26 - 967/576*a^24 + 11/12*a^22 + 7/6*a^20 - 43/18*a^18 + 409/288*a^16 - 37/144*a^14 - 3/32*a^12 + 3/16*a^10 - 17/192*a^8 - 13/48*a^6 + 125/288*a^4 - 35/144*a^2 + 7/144)/a^40)*t^5 + ((1/30*a^76 - 1/40*a^74 + 91/360*a^70 - 35/48*a^68 + 14/15*a^66 - 769/120*a^64 + 245/36*a^62 + 551/16*a^60 - 6007/72*a^58 + 15155/288*a^56 + 1429/80*a^54 - 597/16*a^52 + 1801/48*a^50 - 31907/576*a^48 + 2023/36*a^46 - 2015/72*a^44 - 259/24*a^42 + 26549/720*a^40 - 3419/144*a^38 - 10109/720*a^36 + 5971/240*a^34 + 41/64*a^32 - 7841/360*a^30 + 5405/288*a^28 - 7657/720*a^26 + 1447/160*a^24 - 97/24*a^22 - 41/8*a^20 + 529/72*a^18 - 3451/1152*a^16 - 221/720*a^14 + 247/144*a^12 - 55/24*a^10 + 41/64*a^8 + 91/48*a^6 - 73/32*a^4 + 49/48*a^2 - 967/5760)/a^40)*t^4 + ((1/240*a^86 - 1/720*a^84 + 1/40*a^78 - 1/10*a^76 + 13/180*a^74 + 23/480*a^72 - 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3841/2880*a^132 + 1267/576*a^130 - 143963/120960*a^128 - 21823/4320*a^126 + 176789/30240*a^124 - 21887/1890*a^122 + 140897/2240*a^120 - 409721/6048*a^118 - 5058689/48384*a^116 + 10971727/40320*a^114 - 24697321/120960*a^112 + 6467107/151200*a^110 + 1603559/60480*a^108 - 586763/10080*a^106 + 2841925/24192*a^104 - 4976383/30240*a^102 + 42735743/302400*a^100 - 3159677/60480*a^98 - 509711/30240*a^96 - 1895/72*a^94 + 2725807/17280*a^92 - 146333/756*a^90 + 2756683/120960*a^88 + 5486911/30240*a^86 - 1960811/8640*a^84 + 1469453/8640*a^82 - 2851159/28800*a^80 - 159683/4320*a^78 + 220625/1152*a^76 - 5340803/30240*a^74 + 1986809/72576*a^72 + 2104331/100800*a^70 + 45863/4480*a^68 + 179767/2880*a^66 - 1355147/8640*a^64 + 7132309/60480*a^62 - 2219657/80640*a^60 - 12479/6048*a^58 - 2571157/362880*a^56 - 1620487/120960*a^54 + 12251641/120960*a^52 - 10392133/60480*a^50 + 1565981/16128*a^48 + 313531/4320*a^46 - 7063087/40320*a^44 + 2249171/15120*a^42 - 8618093/151200*a^40 - 287621/30240*a^38 + 3663907/90720*a^36 - 22879/320*a^34 + 8894341/120960*a^32 - 7547/320*a^30 - 2351417/241920*a^28 - 203801/10080*a^26 + 2393357/40320*a^24 - 1966729/40320*a^22 + 1497499/134400*a^20 + 165559/17280*a^18 - 31201/1080*a^16 + 134821/2160*a^14 - 4052323/60480*a^12 + 16619/720*a^10 + 7343/360*a^8 - 272441/10080*a^6 + 268823/20160*a^4 - 6515/2016*a^2 + 16103/50400)/a^88)*t^3 + ((1/1814400*a^200 + 1/362880*a^184 - 1/60480*a^182 + 1/403200*a^180 + 1/120960*a^172 - 1/6720*a^168 + 1/3360*a^166 - 1/362880*a^164 - 1/22680*a^162 + 1/25200*a^160 - 1/2520*a^158 - 1/3360*a^156 + 1/280*a^154 - 577/120960*a^152 - 3/11200*a^150 - 103/103680*a^148 + 1487/181440*a^146 + 283/26880*a^144 - 113/2520*a^142 + 3103/75600*a^140 + 191/3780*a^138 - 2887/40320*a^136 - 1093/4032*a^134 + 16631/24192*a^132 - 87551/75600*a^130 + 1469/3456*a^128 + 70009/30240*a^126 - 22891/16128*a^124 + 1161/224*a^122 - 1696571/67200*a^120 + 477833/24192*a^118 + 9063529/241920*a^116 - 9001483/120960*a^114 + 3819827/80640*a^112 - 113137/10800*a^110 - 16973/20160*a^108 + 144871/30240*a^106 - 272921/20160*a^104 + 1047653/60480*a^102 - 734911/67200*a^100 + 4541/504*a^98 - 749269/40320*a^96 + 28679/864*a^94 - 219811/4320*a^92 + 7875151/151200*a^90 - 147061/10080*a^88 - 524119/15120*a^86 + 3226037/60480*a^84 - 21989/540*a^82 + 351499/33600*a^80 + 143459/4320*a^78 - 724081/12960*a^76 + 1811549/90720*a^74 + 594143/22680*a^72 - 92669/12096*a^70 - 2218987/60480*a^68 + 1793/96*a^66 + 642527/17280*a^64 - 2841/56*a^62 + 10482053/403200*a^60 - 6067/630*a^58 + 407143/120960*a^56 + 76459/3456*a^54 - 457031/6912*a^52 + 2025191/28800*a^50 - 103447/17280*a^48 - 278389/4320*a^46 + 1567357/17280*a^44 - 176809/2160*a^42 + 6116249/129600*a^40 - 76037/25920*a^38 - 625637/25920*a^36 + 17771/720*a^34 - 5347/2880*a^32 - 19601/1080*a^30 - 92471/17280*a^28 + 36113/540*a^26 - 2060213/20160*a^24 + 201973/2880*a^22 - 94879/5760*a^20 - 31903/4032*a^18 + 211703/6720*a^16 - 39929/560*a^14 + 248123/3360*a^12 - 5659/240*a^10 - 853/40*a^8 + 1473/56*a^6 - 61921/5040*a^4 + 7129/2520*a^2 - 671/2520)/a^88)*t^2 + ((1/39916800*a^220 - 1/1814400*a^200 - 1/362880*a^184 + 1/120960*a^182 + 1/3628800*a^180 - 1/120960*a^172 + 1/13440*a^168 - 1/10080*a^166 - 1/60480*a^164 - 1/181440*a^162 - 1/43200*a^160 + 1/5040*a^158 + 1/6720*a^156 - 1/840*a^154 + 3/2240*a^152 + 1/4800*a^150 + 487/725760*a^148 - 491/181440*a^146 - 1133/362880*a^144 + 1/96*a^142 - 43/5760*a^140 - 253/15120*a^138 + 19/1440*a^136 + 671/10080*a^134 - 17753/120960*a^132 + 38917/151200*a^130 - 3503/60480*a^128 - 299/720*a^126 + 22933/241920*a^124 - 161/160*a^122 + 33823/8064*a^120 - 4751/1920*a^118 - 111179/20160*a^116 + 60269/6720*a^114 - 118949/24192*a^112 + 437/504*a^110 + 271/2688*a^108 + 2671/10080*a^106 - 671/3360*a^104 + 313/1080*a^102 - 747179/302400*a^100 + 3407/1344*a^98 + 15803/5040*a^96 - 15461/2160*a^94 + 1759/384*a^92 - 41017/50400*a^90 - 2531/6048*a^88 + 2291/2520*a^86 - 23483/10080*a^84 + 23233/8640*a^82 + 24601/14400*a^80 - 35891/4320*a^78 + 41977/4320*a^76 - 20689/4320*a^74 - 311/270*a^72 + 1093/864*a^70 + 1883/288*a^68 - 8633/864*a^66 - 1907/1728*a^64 + 191/15*a^62 - 36235/3456*a^60 + 1073/540*a^58 + 5309/960*a^56 - 129613/8640*a^54 + 328531/17280*a^52 - 2527/576*a^50 - 71707/4320*a^48 + 13217/720*a^46 - 27581/2880*a^44 + 977/72*a^42 - 77699/4320*a^40 + 5135/864*a^38 + 3659/864*a^36 + 287/48*a^34 - 133/6*a^32 + 2813/144*a^30 + 1903/288*a^28 - 707/18*a^26 + 2491/48*a^24 - 1601/48*a^22 + 767/96*a^20 + 39/16*a^18 - 617/48*a^16 + 361/12*a^14 - 731/24*a^12 + 37/4*a^10 + 17/2*a^8 - 10*a^6 + 9/2*a^4 - a^2 + 1/11)/a^88)*t)*w^11)*z^11 + O(z^12)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dtqsub = dt.subs(z=a^2*z*w)\n",
    "dtqsub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "s*t + ((a^2*s*t^2 - 2*a^2*s*t)*w)*z + (((1/2*a^2*s^2 + (a^4 - a^2)*s)*t^3 + ((1/2*a^6 - 3*a^4 + 1/2*a^2)*s)*t^2 + ((-a^6 + 3*a^4)*s)*t)*w^2 + ((a^2*s^2 - 2*a^2*s)*t^2 + a^2*s*t)*w)*z^2 + (((1/6*s^3 + (1/2*a^4 - 1/2)*s^2 + (a^6 - 3/2*a^4 + 1/2)*s)*t^4 + ((-1/2*a^4 - a^2 + 1/2)*s^2 + (a^8 - 4*a^6 + 2*a^4 + 2*a^2 - 1)*s)*t^3 + ((1/6*a^12 - 3*a^8 + 6*a^6 - 1/2*a^4 - a^2 + 1/3)*s)*t^2 + ((-1/3*a^12 + 3*a^8 - 4*a^6)*s)*t)*w^3 + (-a^4*s*t^3 + (-2*a^2*s^2 + (a^4 + 4*a^2)*s)*t^2 - 2*a^2*s*t)*w^2)*z^3 + (((1/24/a^4*s^4 + ((1/6*a^6 - 1/6)/a^4)*s^3 + ((1/2*a^10 - 1/4*a^8 - 1/2*a^6 + 1/4)/a^4)*s^2 + ((a^12 - 2*a^10 + 1/2*a^8 + 2/3*a^6 - 1/6)/a^4)*s)*t^5 + (((-1/3*a^6 + 1/4*a^4 - 1/2*a^2 + 1/4)/a^4)*s^3 + ((1/4*a^12 - a^10 - 3/4*a^8 + 3/2*a^6 - 3/4*a^4 + 3/2*a^2 - 3/4)/a^4)*s^2 + ((3/2*a^14 - 23/4*a^12 + 9/2*a^10 + 9/4*a^8 - 5/2*a^6 + 3/4*a^4 - 3/2*a^2 + 3/4)/a^4)*s)*t^4 + (((-1/3*a^12 + 1/2*a^10 + 9/8*a^8 - 1/2*a^6 + 5/4*a^4 - 3/2*a^2 + 11/24)/a^4)*s^2 + ((1/3*a^18 + 1/4*a^16 - 6*a^14 + 67/6*a^12 - 3*a^10 - 4*a^8 + 5/3*a^6 - 5/2*a^4 + 3*a^2 - 11/12)/a^4)*s)*t^3 + (((1/24*a^24 - a^18 - 3/4*a^16 + 9*a^14 - 31/3*a^12 + 1/2*a^10 + 9/8*a^8 - 1/6*a^6 + a^4 - a^2 + 1/4)/a^4)*s)*t^2 + ((-1/12*a^20 + a^14 + 3/4*a^12 - 6*a^10 + 5*a^8)*s)*t)*w^4 + ((-1/2*a^4*s^2 + (-a^6 + a^4)*s)*t^4 + (1/2*a^4*s^2 + (-1/2*a^8 + 2*a^6 + 1/2*a^4)*s)*t^3 + ((-a^4 + 3*a^2)*s^2 + (1/2*a^8 - a^6 + 1/2*a^4 - 6*a^2)*s)*t^2 + ((-a^4 + 3*a^2)*s)*t)*w^3 + ((1/2*s^3 - 3/2*s^2 + 3/2*s)*t^3 + ((1/2*a^4 - a^2 + 3/2)*s^2 + (-a^4 + 2*a^2 - 3)*s)*t^2 + ((1/2*a^4 - a^2 + 1)*s)*t)*w^2)*z^4 + (((1/120/a^10*s^5 + ((1/24*a^8 - 1/24)/a^10)*s^4 + ((1/6*a^14 - 1/12*a^12 - 1/6*a^8 + 1/12)/a^10)*s^3 + ((1/2*a^18 - 1/2*a^16 - 1/2*a^14 + 1/3*a^12 + 1/4*a^8 - 1/12)/a^10)*s^2 + ((a^20 - 5/2*a^18 + 5/4*a^16 + 5/6*a^14 - 5/12*a^12 - 5/24*a^8 + 1/24)/a^10)*s)*t^6 + (((-1/12*a^8 + 1/12*a^4 - 1/6*a^2 + 1/12)/a^10)*s^4 + ((1/12*a^16 - 1/2*a^14 + 1/3*a^12 - 1/2*a^10 + 7/12*a^8 - 1/3*a^4 + 2/3*a^2 - 1/3)/a^10)*s^3 + ((1/2*a^20 - 3/2*a^18 - 3/4*a^16 + 3*a^14 - 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9/4*a^8 + 3/2*a^4 - a^2 + 1/5)/a^10)*s)*t^2 + ((-1/60*a^30 + 1/4*a^22 + 1/2*a^18 - 2*a^16 - 3*a^14 + 10*a^12 - 6*a^10)*s)*t)*w^5 + ((-1/6*a^2*s^3 + (-1/2*a^6 + 1/2*a^2)*s^2 + (-a^8 + 3/2*a^6 - 1/2*a^2)*s)*t^5 + (1/6*a^2*s^3 + (2*a^4 - a^2)*s^2 + (-a^10 + 3*a^8 + 1/2*a^6 - 4*a^4 + 3/2*a^2)*s)*t^4 + ((1/2*a^6 - 2*a^4 + 1/2*a^2)*s^2 + (-1/6*a^14 + 2*a^10 - 2*a^8 - 7/2*a^6 + 3*a^4 - 4/3*a^2)*s)*t^3 + ((-1/3*a^8 + 3*a^4 - 4*a^2)*s^2 + (1/6*a^14 - a^10 + 2/3*a^8 + 3/2*a^6 - 5*a^4 + 25/3*a^2)*s)*t^2 + ((-1/3*a^8 + 3*a^4 - 4*a^2)*s)*t)*w^4 + (1/2*a^2*s*t^4 + (-1/a^2*s^3 + 3/a^2*s^2 + ((-1/2*a^8 + a^6 - 3/2*a^4 - 3)/a^2)*s)*t^3 + (((-a^4 + 2*a^2 - 3)/a^2)*s^2 + ((1/2*a^8 - a^6 + 3*a^4 - 4*a^2 + 6)/a^2)*s)*t^2 + (((-a^4 + 2*a^2 - 2)/a^2)*s)*t)*w^3)*z^5 + (((1/720/a^18*s^6 + ((1/120*a^10 - 1/120)/a^18)*s^5 + ((1/24*a^18 - 1/48*a^16 - 1/24*a^10 + 1/48)/a^18)*s^4 + ((1/6*a^24 - 1/6*a^22 - 5/36*a^18 + 1/12*a^16 + 1/12*a^10 - 1/36)/a^18)*s^3 + ((1/2*a^28 - 3/4*a^26 - 3/8*a^24 + 2/3*a^22 + 1/6*a^18 - 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6)/a^26)*s)*t)*w^8 + ((1/144/a^16*s^3 + ((1/48*a^4 - 1/48)/a^16)*s^2 + ((1/24*a^6 - 1/16*a^4 + 1/48)/a^16)*s)*t^8 + (((-1/24*a^4 + 1/12*a^2 - 5/72)/a^16)*s^3 + ((-1/8*a^12 + 1/4*a^10 - 1/16*a^8 - 7/24*a^6 + 11/48*a^4 - 1/24)/a^20)*s^2 + ((-1/4*a^14 + 11/12*a^12 - 5/4*a^10 + 1/2*a^8 + 1/3*a^6 - 1/4*a^4 - 1/12*a^2 + 1/12)/a^20)*s)*t^7 + (((1/36*a^12 - 7/48*a^8 + 1/12*a^6 + 1/3*a^4 - 1/2*a^2 + 35/144)/a^16)*s^3 + ((1/12*a^20 - 25/48*a^16 + 1/4*a^14 + 21/16*a^12 - 5/4*a^10 - 53/48*a^8 + 13/6*a^6 - 11/16*a^4 - 1/2*a^2 + 5/12)/a^20)*s^2 + ((1/6*a^28 - 1/4*a^26 - 7/8*a^24 + 119/72*a^22 + 9/4*a^20 - 379/48*a^18 + 79/12*a^16 + 1/12*a^14 - 19/8*a^12 - 55/144*a^10 + 23/12*a^8 - 7/8*a^6 - 1/24*a^2 + 1/8)/a^26)*s)*t^6 + (((-1/72*a^6 + 1/8*a^2 - 1/6)/a^34)*s^5 + ((5/72*a^6 - 5/8*a^2 + 5/6)/a^34)*s^4 + ((-1/144*a^42 + 1/18*a^36 + 1/24*a^34 - 1/4*a^32 + 1/12*a^30 + 7/16*a^26 - 1/4*a^24 - 17/24*a^22 + 11/12*a^20 - 25/72*a^18 - 5/36*a^6 + 5/4*a^2 - 5/3)/a^34)*s^3 + ((-1/48*a^46 + 1/48*a^42 + 1/6*a^40 + 1/8*a^38 - 11/12*a^36 + 5/24*a^34 + 7/12*a^32 + 17/24*a^30 + 3/8*a^28 - 85/24*a^26 + 1/4*a^24 + 107/16*a^22 - 149/24*a^20 - 11/48*a^18 + 3*a^16 - 35/24*a^14 + 5/36*a^6 - 5/4*a^2 + 5/3)/a^34)*s^2 + ((-1/24*a^48 + 1/16*a^46 + 5/16*a^42 - 1/4*a^40 - 41/24*a^38 + 31/12*a^36 - 37/24*a^34 + 109/24*a^32 - 601/144*a^30 - 71/6*a^28 + 1135/48*a^26 - 26/3*a^24 - 247/24*a^22 + 21/4*a^20 + 325/36*a^18 - 41/4*a^16 + 43/12*a^14 - 5/4*a^12 + 23/12*a^10 - 5/4*a^8 - 5/72*a^6 + 5/8*a^2 - 5/6)/a^34)*s)*t^5 + (((-1/12*a^10 + 1/6*a^8 + 11/18*a^6 - 5/2*a^4 + 13/4*a^2 - 5/3)/a^34)*s^4 + ((1/144*a^42 - 1/18*a^36 - 1/24*a^34 + 1/4*a^32 - 1/9*a^30 - 7/24*a^26 + 1/6*a^24 + 5/12*a^22 - 1/2*a^20 + 1/6*a^18 + 1/3*a^10 - 2/3*a^8 - 22/9*a^6 + 10*a^4 - 13*a^2 + 20/3)/a^34)*s^3 + ((1/24*a^44 - 1/24*a^42 - 7/24*a^38 + 1/12*a^36 + 5/3*a^34 - 7/3*a^32 + 37/48*a^30 - 11/8*a^28 + 37/16*a^26 + 17/4*a^24 - 289/24*a^22 + 47/6*a^20 + 65/24*a^18 - 11/2*a^16 + 25/12*a^14 - 1/2*a^10 + a^8 + 11/3*a^6 - 15*a^4 + 39/2*a^2 - 10)/a^34)*s^2 + ((-1/24*a^50 + 1/8*a^48 - 1/16*a^46 + 1/4*a^44 - 95/144*a^42 - 5/3*a^40 + 271/48*a^38 - 11/3*a^36 + 7/8*a^34 - 57/8*a^32 + 233/72*a^30 + 545/24*a^28 - 231/8*a^26 - 149/24*a^24 + 1313/48*a^22 - 9/8*a^20 - 1801/72*a^18 + 431/24*a^16 - 49/8*a^14 + 9*a^12 - 81/8*a^10 + 89/24*a^8 - 22/9*a^6 + 10*a^4 - 13*a^2 + 20/3)/a^34)*s)*t^4 + (((-1/18*a^18 + 19/24*a^14 - 5/6*a^12 - 79/24*a^10 + 6*a^8 + 253/72*a^6 - 17*a^4 + 131/8*a^2 - 35/6)/a^34)*s^3 + ((1/48*a^46 - 1/24*a^44 + 1/48*a^42 - 1/6*a^40 + 1/6*a^38 + 5/6*a^36 - 47/24*a^34 + 7/4*a^32 - 23/24*a^30 + 3/4*a^28 + 1/24*a^26 - 7/2*a^24 + 13/2*a^22 - 7/2*a^20 - 11/6*a^18 + 3*a^16 - 27/8*a^14 + 5/2*a^12 + 79/8*a^10 - 18*a^8 - 253/24*a^6 + 51*a^4 - 393/8*a^2 + 35/2)/a^34)*s^2 + ((-1/144*a^54 + 1/12*a^50 - 5/72*a^48 + 1/48*a^46 - 19/24*a^44 + 41/72*a^42 + 15/4*a^40 - 287/48*a^38 + 7/36*a^36 + 9/8*a^34 + 85/12*a^32 - 317/144*a^30 - 145/8*a^28 + 119/8*a^26 + 227/12*a^24 - 85/3*a^22 - 61/12*a^20 + 1717/72*a^18 - 199/24*a^16 + 149/24*a^14 - 83/4*a^12 + 209/24*a^10 + 47/4*a^8 + 253/24*a^6 - 51*a^4 + 393/8*a^2 - 35/2)/a^34)*s)*t^3 + (((-1/72*a^30 + 1/8*a^26 - 1/18*a^24 + 1/12*a^22 - 3/2*a^20 + 3/4*a^18 + 11/2*a^16 - 53/8*a^14 + 3/2*a^12 - 223/24*a^10 + 101/6*a^8 + 109/18*a^6 - 67/2*a^4 + 113/4*a^2 - 25/3)/a^34)*s^2 + ((1/144*a^54 - 1/24*a^50 - 1/72*a^48 - 1/48*a^46 + 13/24*a^44 - 2/9*a^42 - 11/6*a^40 + 49/24*a^38 + 13/18*a^36 - 5/24*a^34 - 29/8*a^32 + 109/72*a^30 + 21/4*a^28 - 35/12*a^26 - 335/36*a^24 + 127/12*a^22 + 6*a^20 - 107/12*a^18 - 49/4*a^16 + 51/4*a^14 + 15/2*a^12 + 103/12*a^10 - 92/3*a^8 - 109/9*a^6 + 67*a^4 - 113/2*a^2 + 50/3)/a^34)*s)*t^2 + (((-1/72*a^30 + 1/8*a^26 - 1/18*a^24 + 1/12*a^22 - 3/2*a^20 + 29/36*a^18 + 11/2*a^16 - 89/12*a^14 + 7/3*a^12 - 73/12*a^10 + 11*a^8 + 19/6*a^6 - 19*a^4 + 15*a^2 - 4)/a^34)*s)*t)*w^7 + (-1/120/a^28*s*t^7 + (-1/60/a^38*s^6 + 1/10/a^38*s^5 - 1/4/a^38*s^4 + 1/3/a^38*s^3 - 1/4/a^38*s^2 + ((1/12*a^14 - 1/6*a^12 + 1/8*a^10 + 1/10)/a^38)*s)*t^6 + (((-1/6*a^4 + 1/3*a^2 - 1/4)/a^38)*s^5 + ((5/6*a^4 - 5/3*a^2 + 5/4)/a^38)*s^4 + ((-5/3*a^4 + 10/3*a^2 - 5/2)/a^38)*s^3 + ((5/3*a^4 - 10/3*a^2 + 5/2)/a^38)*s^2 + ((-1/12*a^22 + 3/8*a^18 - 4/3*a^14 + 5/3*a^12 - 17/24*a^10 - 5/6*a^4 + 5/3*a^2 - 5/4)/a^38)*s)*t^5 + (((-1/6*a^12 + 3/4*a^8 - 8/3*a^4 + 10/3*a^2 - 17/12)/a^38)*s^4 + ((2/3*a^12 - 3*a^8 + 32/3*a^4 - 40/3*a^2 + 17/3)/a^38)*s^3 + ((-a^12 + 9/2*a^8 - 16*a^4 + 20*a^2 - 17/2)/a^38)*s^2 + ((1/24*a^34 - 1/3*a^28 - 1/6*a^26 + 4/3*a^24 - a^22 + 3/2*a^20 - 13/4*a^18 + 71/12*a^14 - 31/6*a^12 + 15/8*a^10 - 3*a^8 + 32/3*a^4 - 40/3*a^2 + 17/3)/a^38)*s)*t^4 + (((-1/12*a^24 + 2/3*a^18 + 1/3*a^16 - 8/3*a^14 + 2*a^12 - 3*a^10 + 13/2*a^8 - 71/6*a^4 + 35/3*a^2 - 15/4)/a^38)*s^3 + ((1/4*a^24 - 2*a^18 - a^16 + 8*a^14 - 6*a^12 + 9*a^10 - 39/2*a^8 + 71/2*a^4 - 35*a^2 + 45/4)/a^38)*s^2 + ((-1/120*a^50 + 1/12*a^42 + 1/6*a^38 - 1/2*a^36 - 7/8*a^34 + 2*a^32 - a^30 + a^28 + 1/2*a^26 - 17/4*a^24 + 43/12*a^22 - 9/2*a^20 + 73/8*a^18 + a^16 - 53/3*a^14 + 43/3*a^12 - 677/60*a^10 + 39/2*a^8 - 71/2*a^4 + 35*a^2 - 45/4)/a^38)*s)*t^3 + (((-1/60*a^40 + 1/6*a^32 + 1/3*a^28 - a^26 - 7/4*a^24 + 4*a^22 - 2*a^20 + 2*a^18 + a^16 - 8*a^14 + 43/6*a^12 - 9*a^10 + 57/4*a^8 - 58/3*a^4 + 50/3*a^2 - 137/30)/a^38)*s^2 + ((1/120*a^50 - 1/12*a^42 + 1/30*a^40 - 1/6*a^38 + 1/2*a^36 + 5/6*a^34 - 7/3*a^32 + a^30 - 4/3*a^28 + 5/3*a^26 + 37/6*a^24 - 21/2*a^22 + 7*a^20 - 33/4*a^18 - 2*a^16 + 21*a^14 - 55/3*a^12 + 19*a^10 - 57/2*a^8 + 116/3*a^4 - 100/3*a^2 + 137/15)/a^38)*s)*t^2 + (((-1/60*a^40 + 1/6*a^32 + 1/3*a^28 - a^26 - 5/3*a^24 + 4*a^22 - 2*a^20 + 4/3*a^18 + 2/3*a^16 - 16/3*a^14 + 5*a^12 - 6*a^10 + 17/2*a^8 - 10*a^4 + 8*a^2 - 2)/a^38)*s)*t)*w^6)*z^11 + O(z^12)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "part1 = num2 + dtsqsub*num1\n",
    "part1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[s*t,\n",
       " (2*a^2*s*t^2 - 2*a^2*s*t)*w,\n",
       " ((a^2*s^2 + (6*a^4 - 3*a^2)*s)*t^3 + ((2*a^6 - 12*a^4 + 2*a^2)*s)*t^2 + ((-2*a^6 + 6*a^4)*s)*t)*w^2 + ((2*a^2*s^2 - 4*a^2*s)*t^2 + 2*a^2*s*t)*w,\n",
       " ((s^3 + (6*a^4 - 3)*s^2 + (24*a^6 - 24*a^4 + 4)*s)*t^4 + ((-3*a^4 - 6*a^2 + 3)*s^2 + (18*a^8 - 72*a^6 + 27*a^4 + 18*a^2 - 9)*s)*t^3 + ((2*a^12 - 36*a^8 + 72*a^6 - 6*a^4 - 12*a^2 + 4)*s)*t^2 + ((-2*a^12 + 18*a^8 - 24*a^6)*s)*t)*w^3 + ((6*a^4*s^2 - 18*a^4*s)*t^3 + (-12*a^2*s^2 + (12*a^4 + 24*a^2)*s)*t^2 - 12*a^2*s*t)*w^2,\n",
       " ((1/a^4*s^4 + ((8*a^6 - 4)/a^4)*s^3 + ((36*a^10 - 12*a^8 - 24*a^6 + 6)/a^4)*s^2 + ((120*a^12 - 180*a^10 + 30*a^8 + 40*a^6 - 5)/a^4)*s)*t^5 + (((-8*a^6 + 6*a^4 - 12*a^2 + 6)/a^4)*s^3 + ((12*a^12 - 48*a^10 - 36*a^8 + 48*a^6 - 18*a^4 + 36*a^2 - 18)/a^4)*s^2 + ((144*a^14 - 528*a^12 + 336*a^10 + 144*a^8 - 128*a^6 + 24*a^4 - 48*a^2 + 24)/a^4)*s)*t^4 + (((-8*a^12 + 12*a^10 + 27*a^8 - 12*a^6 + 30*a^4 - 36*a^2 + 11)/a^4)*s^2 + ((24*a^18 + 18*a^16 - 432*a^14 + 780*a^12 - 180*a^10 - 207*a^8 + 84*a^6 - 90*a^4 + 108*a^2 - 33)/a^4)*s)*t^3 + (((2*a^24 - 48*a^18 - 36*a^16 + 432*a^14 - 496*a^12 + 24*a^10 + 54*a^8 - 8*a^6 + 48*a^4 - 48*a^2 + 12)/a^4)*s)*t^2 + ((-2*a^20 + 24*a^14 + 18*a^12 - 144*a^10 + 120*a^8)*s)*t)*w^4 + (((24*a^6 - 24*a^4)*s^2 + (-96*a^6 + 48*a^4)*s)*t^4 + ((12*a^8 - 24*a^6 - 24*a^4)*s^2 + (-36*a^8 + 144*a^6 + 72*a^4)*s)*t^3 + ((-24*a^4 + 72*a^2)*s^2 + (24*a^8 - 48*a^6 - 24*a^4 - 144*a^2)*s)*t^2 + ((-24*a^4 + 72*a^2)*s)*t)*w^3 + ((12*s^3 - 36*s^2 + 36*s)*t^3 + ((12*a^4 - 24*a^2 + 36)*s^2 + (-24*a^4 + 48*a^2 - 72)*s)*t^2 + ((12*a^4 - 24*a^2 + 24)*s)*t)*w^2]"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# CGF of labeled digraphs,\n",
    "# with the marking variable t for strongly connected components\n",
    "# and the marking variable s for source-like components (we take a parameter a = sqrt{2})\n",
    "qdts = part1*dtqsub\n",
    "QDTS = [qdts[i] * i.factorial() for i in srange(disp)]\n",
    "QDTS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "disp = 6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[s*t,\n",
       " (2*a^2*s*t^2 - 2*a^2*s*t)*w,\n",
       " ((a^2*s^2 + (6*a^4 - 3*a^2)*s)*t^3 + ((2*a^6 - 12*a^4 + 2*a^2)*s)*t^2 + ((-2*a^6 + 6*a^4)*s)*t)*w^2 + ((2*a^2*s^2 - 4*a^2*s)*t^2 + 2*a^2*s*t)*w,\n",
       " ((s^3 + (6*a^4 - 3)*s^2 + (24*a^6 - 24*a^4 + 4)*s)*t^4 + ((-3*a^4 - 6*a^2 + 3)*s^2 + (18*a^8 - 72*a^6 + 27*a^4 + 18*a^2 - 9)*s)*t^3 + ((2*a^12 - 36*a^8 + 72*a^6 - 6*a^4 - 12*a^2 + 4)*s)*t^2 + ((-2*a^12 + 18*a^8 - 24*a^6)*s)*t)*w^3 + ((6*a^4*s^2 - 18*a^4*s)*t^3 + (-12*a^2*s^2 + (12*a^4 + 24*a^2)*s)*t^2 - 12*a^2*s*t)*w^2,\n",
       " ((1/a^4*s^4 + ((8*a^6 - 4)/a^4)*s^3 + ((36*a^10 - 12*a^8 - 24*a^6 + 6)/a^4)*s^2 + ((120*a^12 - 180*a^10 + 30*a^8 + 40*a^6 - 5)/a^4)*s)*t^5 + (((-8*a^6 + 6*a^4 - 12*a^2 + 6)/a^4)*s^3 + ((12*a^12 - 48*a^10 - 36*a^8 + 48*a^6 - 18*a^4 + 36*a^2 - 18)/a^4)*s^2 + ((144*a^14 - 528*a^12 + 336*a^10 + 144*a^8 - 128*a^6 + 24*a^4 - 48*a^2 + 24)/a^4)*s)*t^4 + (((-8*a^12 + 12*a^10 + 27*a^8 - 12*a^6 + 30*a^4 - 36*a^2 + 11)/a^4)*s^2 + ((24*a^18 + 18*a^16 - 432*a^14 + 780*a^12 - 180*a^10 - 207*a^8 + 84*a^6 - 90*a^4 + 108*a^2 - 33)/a^4)*s)*t^3 + (((2*a^24 - 48*a^18 - 36*a^16 + 432*a^14 - 496*a^12 + 24*a^10 + 54*a^8 - 8*a^6 + 48*a^4 - 48*a^2 + 12)/a^4)*s)*t^2 + ((-2*a^20 + 24*a^14 + 18*a^12 - 144*a^10 + 120*a^8)*s)*t)*w^4 + (((24*a^6 - 24*a^4)*s^2 + (-96*a^6 + 48*a^4)*s)*t^4 + ((12*a^8 - 24*a^6 - 24*a^4)*s^2 + (-36*a^8 + 144*a^6 + 72*a^4)*s)*t^3 + ((-24*a^4 + 72*a^2)*s^2 + (24*a^8 - 48*a^6 - 24*a^4 - 144*a^2)*s)*t^2 + ((-24*a^4 + 72*a^2)*s)*t)*w^3 + ((12*s^3 - 36*s^2 + 36*s)*t^3 + ((12*a^4 - 24*a^2 + 36)*s^2 + (-24*a^4 + 48*a^2 - 72)*s)*t^2 + ((12*a^4 - 24*a^2 + 24)*s)*t)*w^2,\n",
       " ((1/a^10*s^5 + ((10*a^8 - 5)/a^10)*s^4 + ((60*a^14 - 20*a^12 - 40*a^8 + 10)/a^10)*s^3 + ((240*a^18 - 180*a^16 - 180*a^14 + 80*a^12 + 60*a^8 - 10)/a^10)*s^2 + ((720*a^20 - 1440*a^18 + 540*a^16 + 360*a^14 - 120*a^12 - 60*a^8 + 6)/a^10)*s)*t^6 + (((-10*a^8 + 10*a^4 - 20*a^2 + 10)/a^10)*s^4 + ((20*a^16 - 120*a^14 + 80*a^12 - 120*a^10 + 100*a^8 - 40*a^4 + 80*a^2 - 40)/a^10)*s^3 + ((180*a^20 - 540*a^18 - 240*a^16 + 780*a^14 - 360*a^12 + 360*a^10 - 240*a^8 + 60*a^4 - 120*a^2 + 60)/a^10)*s^2 + ((1200*a^22 - 4500*a^20 + 3900*a^18 + 1000*a^16 - 2100*a^14 + 700*a^12 - 600*a^10 + 350*a^8 - 50*a^4 + 100*a^2 - 50)/a^10)*s)*t^5 + (((-20*a^16 + 60*a^14 - 50*a^12 + 120*a^10 - 105*a^8 + 120*a^4 - 120*a^2 + 35)/a^10)*s^3 + ((20*a^24 - 260*a^20 + 240*a^18 + 630*a^16 - 660*a^14 + 550*a^12 - 720*a^10 + 425*a^8 - 360*a^4 + 360*a^2 - 105)/a^10)*s^2 + ((240*a^26 + 280*a^24 - 4800*a^22 + 8960*a^20 - 3120*a^18 - 3440*a^16 + 2640*a^14 - 1800*a^12 + 1920*a^10 - 860*a^8 + 480*a^4 - 480*a^2 + 140)/a^10)*s)*t^4 + (((-15*a^24 + 80*a^20 + 20*a^18 - 230*a^16 + 220*a^14 - 450*a^12 + 540*a^10 - 365*a^8 + 290*a^4 - 220*a^2 + 50)/a^10)*s^2 + ((30*a^32 + 60*a^28 - 720*a^26 - 975*a^24 + 7200*a^22 - 8220*a^20 + 660*a^18 + 2340*a^16 - 1380*a^14 + 2190*a^12 - 2340*a^10 + 1275*a^8 - 870*a^4 + 660*a^2 - 150)/a^10)*s)*t^3 + (((2*a^40 - 60*a^32 - 120*a^28 + 720*a^26 + 1050*a^24 - 4800*a^22 + 3760*a^20 + 40*a^18 - 420*a^16 + 320*a^14 - 800*a^12 + 840*a^10 - 540*a^8 + 360*a^4 - 240*a^2 + 48)/a^10)*s)*t^2 + ((-2*a^30 + 30*a^22 + 60*a^18 - 240*a^16 - 360*a^14 + 1200*a^12 - 720*a^10)*s)*t)*w^5 + ((-20*a^2*s^3 + (120*a^8 - 240*a^6 + 80*a^2)*s^2 + (-600*a^8 + 600*a^6 - 100*a^2)*s)*t^5 + (20*a^2*s^3 + (120*a^10 - 240*a^8 - 120*a^6 + 480*a^4 - 180*a^2)*s^2 + (-480*a^10 + 1440*a^8 + 240*a^6 - 960*a^4 + 320*a^2)*s)*t^4 + ((20*a^14 - 120*a^10 + 240*a^6 - 120*a^4 + 100*a^2)*s^2 + (-60*a^14 + 720*a^10 - 720*a^8 - 1080*a^6 + 360*a^4 - 300*a^2)*s)*t^3 + ((-40*a^8 + 360*a^4 - 480*a^2)*s^2 + (40*a^14 - 240*a^10 + 80*a^8 + 360*a^6 - 480*a^4 + 1040*a^2)*s)*t^2 + ((-40*a^8 + 360*a^4 - 480*a^2)*s)*t)*w^4 + ((60*a^2*s^3 - 180*a^2*s^2 + 240*a^2*s)*t^4 + (-120/a^2*s^3 + ((60*a^8 - 120*a^6 + 180*a^4 + 360)/a^2)*s^2 + ((-180*a^8 + 360*a^6 - 540*a^4 - 360)/a^2)*s)*t^3 + (((-120*a^4 + 240*a^2 - 360)/a^2)*s^2 + ((120*a^8 - 240*a^6 + 480*a^4 - 480*a^2 + 720)/a^2)*s)*t^2 + (((-120*a^4 + 240*a^2 - 240)/a^2)*s)*t)*w^3]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QDTS = [qdts[i] * i.factorial() for i in srange(disp)]\n",
    "QDTS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[s*t,\n",
       " 4*s*t^2*w - 4*s*t*w,\n",
       " 2*s^2*t^3*w^2 + 18*s*t^3*w^2 + 4*s^2*t^2*w - 28*s*t^2*w^2 - 8*s*t^2*w + 8*s*t*w^2 + 4*s*t*w,\n",
       " s^3*t^4*w^3 + 21*s^2*t^4*w^3 - 21*s^2*t^3*w^3 + 100*s*t^4*w^3 + 24*s^2*t^3*w^2 - 153*s*t^3*w^3 - 24*s^2*t^2*w^2 - 72*s*t^3*w^2 + 84*s*t^2*w^3 + 96*s*t^2*w^2 - 32*s*t*w^3 - 24*s*t*w^2,\n",
       " 1/4*s^4*t^5*w^4 + 15*s^3*t^5*w^4 - 29/2*s^3*t^4*w^4 + 387/2*s^2*t^5*w^4 - 489/2*s^2*t^4*w^4 + 2715/4*s*t^5*w^4 + 96*s^2*t^4*w^3 + 267/4*s^2*t^3*w^4 - 826*s*t^4*w^4 + 12*s^3*t^3*w^2 - 96*s^2*t^3*w^3 - 576*s*t^4*w^3 + 2943/4*s*t^3*w^4 - 36*s^2*t^3*w^2 + 48*s^2*t^2*w^3 + 864*s*t^3*w^3 - 93*s*t^2*w^4 + 36*s^2*t^2*w^2 + 36*s*t^3*w^2 - 384*s*t^2*w^3 - 512*s*t*w^4 - 72*s*t^2*w^2 + 48*s*t*w^3 + 24*s*t*w^2,\n",
       " 1/32*s^5*t^6*w^5 + 155/32*s^4*t^6*w^5 - 75/16*s^4*t^5*w^5 + 2885/16*s^3*t^6*w^5 - 925/4*s^3*t^5*w^5 + 29915/16*s^2*t^6*w^5 - 40*s^3*t^5*w^4 + 1795/32*s^3*t^4*w^5 - 17265/8*s^2*t^5*w^5 + 87843/16*s*t^6*w^5 + 40*s^3*t^4*w^4 + 160*s^2*t^5*w^4 + 33495/32*s^2*t^4*w^5 - 67625/16*s*t^5*w^5 + 120*s^3*t^4*w^3 + 600*s^2*t^4*w^4 - 5000*s*t^5*w^4 - 8295/16*s^2*t^3*w^5 + 62755/8*s*t^4*w^5 - 60*s^3*t^3*w^3 - 360*s^2*t^4*w^3 + 360*s^2*t^3*w^4 + 6400*s*t^4*w^4 + 114645/16*s*t^3*w^5 + 540*s^2*t^3*w^3 + 480*s*t^4*w^3 - 160*s^2*t^2*w^4 - 3960*s*t^3*w^4 + 21235/2*s*t^2*w^5 - 180*s^2*t^2*w^3 - 1260*s*t^3*w^3 + 1760*s*t^2*w^4 - 27136*s*t*w^5 + 840*s*t^2*w^3 - 160*s*t*w^4 - 120*s*t*w^3]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# CGF of labeled digraphs,\n",
    "# with the marking variable t for strongly connected components\n",
    "# and the marking variable s for source-like components\n",
    "QDTSsub = [sum(sum(sum(QDTS[i][k][j][l].subs(a=sqrt(2)) * s^l * t^j * w^k for l in srange(N)) for j in srange(N)) for k in srange(N)) for i in srange(disp)]\n",
    "QDTSsub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[t,\n",
       " 4*t^2*w - 4*t*w,\n",
       " 20*t^3*w^2 - 28*t^2*w^2 - 4*t^2*w + 8*t*w^2 + 4*t*w,\n",
       " 122*t^4*w^3 - 174*t^3*w^3 - 48*t^3*w^2 + 84*t^2*w^3 + 72*t^2*w^2 - 32*t*w^3 - 24*t*w^2,\n",
       " 1775/2*t^5*w^4 - 1085*t^4*w^4 - 480*t^4*w^3 + 1605/2*t^3*w^4 + 768*t^3*w^3 - 93*t^2*w^4 + 12*t^3*w^2 - 336*t^2*w^3 - 512*t*w^4 - 36*t^2*w^2 + 48*t*w^3 + 24*t*w^2,\n",
       " 120721/16*t^6*w^5 - 52965/8*t^5*w^5 - 4880*t^5*w^4 + 143155/16*t^4*w^5 + 7040*t^4*w^4 + 53175/8*t^3*w^5 + 240*t^4*w^3 - 3600*t^3*w^4 + 21235/2*t^2*w^5 - 780*t^3*w^3 + 1600*t^2*w^4 - 27136*t*w^5 + 660*t^2*w^3 - 160*t*w^4 - 120*t*w^3]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# CGF of labeled digraphs,\n",
    "# with the marking variable t for strongly connected components\n",
    "QDTSsubsub = [QDTSsub[i].subs(s=1) for i in srange(disp)]\n",
    "QDTSsubsub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[s*t,\n",
       " (2*a^2*s*t^2 - 2*a^2*s*t)*w,\n",
       " ((1/2*a^4*s^2 + (3*a^6 - 3/2*a^4)*s)*t^3 + ((a^8 - 6*a^6 + a^4)*s)*t^2 + ((-a^8 + 3*a^6)*s)*t)*w^2 + ((a^4*s^2 - 2*a^4*s)*t^2 + a^4*s*t)*w,\n",
       " ((1/6*a^6*s^3 + (a^10 - 1/2*a^6)*s^2 + (4*a^12 - 4*a^10 + 2/3*a^6)*s)*t^4 + ((-1/2*a^10 - a^8 + 1/2*a^6)*s^2 + (3*a^14 - 12*a^12 + 9/2*a^10 + 3*a^8 - 3/2*a^6)*s)*t^3 + ((1/3*a^18 - 6*a^14 + 12*a^12 - a^10 - 2*a^8 + 2/3*a^6)*s)*t^2 + ((-1/3*a^18 + 3*a^14 - 4*a^12)*s)*t)*w^3 + ((a^10*s^2 - 3*a^10*s)*t^3 + (-2*a^8*s^2 + (2*a^10 + 4*a^8)*s)*t^2 - 2*a^8*s*t)*w^2,\n",
       " ((1/24*a^8*s^4 + (1/3*a^14 - 1/6*a^8)*s^3 + (3/2*a^18 - 1/2*a^16 - a^14 + 1/4*a^8)*s^2 + (5*a^20 - 15/2*a^18 + 5/4*a^16 + 5/3*a^14 - 5/24*a^8)*s)*t^5 + ((-1/3*a^14 + 1/4*a^12 - 1/2*a^10 + 1/4*a^8)*s^3 + (1/2*a^20 - 2*a^18 - 3/2*a^16 + 2*a^14 - 3/4*a^12 + 3/2*a^10 - 3/4*a^8)*s^2 + (6*a^22 - 22*a^20 + 14*a^18 + 6*a^16 - 16/3*a^14 + a^12 - 2*a^10 + a^8)*s)*t^4 + ((-1/3*a^20 + 1/2*a^18 + 9/8*a^16 - 1/2*a^14 + 5/4*a^12 - 3/2*a^10 + 11/24*a^8)*s^2 + (a^26 + 3/4*a^24 - 18*a^22 + 65/2*a^20 - 15/2*a^18 - 69/8*a^16 + 7/2*a^14 - 15/4*a^12 + 9/2*a^10 - 11/8*a^8)*s)*t^3 + ((1/12*a^32 - 2*a^26 - 3/2*a^24 + 18*a^22 - 62/3*a^20 + a^18 + 9/4*a^16 - 1/3*a^14 + 2*a^12 - 2*a^10 + 1/2*a^8)*s)*t^2 + ((-1/12*a^32 + a^26 + 3/4*a^24 - 6*a^22 + 5*a^20)*s)*t)*w^4 + (((a^18 - a^16)*s^2 + (-4*a^18 + 2*a^16)*s)*t^4 + ((1/2*a^20 - a^18 - a^16)*s^2 + (-3/2*a^20 + 6*a^18 + 3*a^16)*s)*t^3 + ((-a^16 + 3*a^14)*s^2 + (a^20 - 2*a^18 - a^16 - 6*a^14)*s)*t^2 + ((-a^16 + 3*a^14)*s)*t)*w^3 + ((1/2*a^12*s^3 - 3/2*a^12*s^2 + 3/2*a^12*s)*t^3 + ((1/2*a^16 - a^14 + 3/2*a^12)*s^2 + (-a^16 + 2*a^14 - 3*a^12)*s)*t^2 + ((1/2*a^16 - a^14 + a^12)*s)*t)*w^2,\n",
       " ((1/120*a^10*s^5 + (1/12*a^18 - 1/24*a^10)*s^4 + (1/2*a^24 - 1/6*a^22 - 1/3*a^18 + 1/12*a^10)*s^3 + (2*a^28 - 3/2*a^26 - 3/2*a^24 + 2/3*a^22 + 1/2*a^18 - 1/12*a^10)*s^2 + (6*a^30 - 12*a^28 + 9/2*a^26 + 3*a^24 - a^22 - 1/2*a^18 + 1/20*a^10)*s)*t^6 + ((-1/12*a^18 + 1/12*a^14 - 1/6*a^12 + 1/12*a^10)*s^4 + (1/6*a^26 - a^24 + 2/3*a^22 - a^20 + 5/6*a^18 - 1/3*a^14 + 2/3*a^12 - 1/3*a^10)*s^3 + (3/2*a^30 - 9/2*a^28 - 2*a^26 + 13/2*a^24 - 3*a^22 + 3*a^20 - 2*a^18 + 1/2*a^14 - a^12 + 1/2*a^10)*s^2 + (10*a^32 - 75/2*a^30 + 65/2*a^28 + 25/3*a^26 - 35/2*a^24 + 35/6*a^22 - 5*a^20 + 35/12*a^18 - 5/12*a^14 + 5/6*a^12 - 5/12*a^10)*s)*t^5 + ((-1/6*a^26 + 1/2*a^24 - 5/12*a^22 + a^20 - 7/8*a^18 + a^14 - a^12 + 7/24*a^10)*s^3 + (1/6*a^34 - 13/6*a^30 + 2*a^28 + 21/4*a^26 - 11/2*a^24 + 55/12*a^22 - 6*a^20 + 85/24*a^18 - 3*a^14 + 3*a^12 - 7/8*a^10)*s^2 + (2*a^36 + 7/3*a^34 - 40*a^32 + 224/3*a^30 - 26*a^28 - 86/3*a^26 + 22*a^24 - 15*a^22 + 16*a^20 - 43/6*a^18 + 4*a^14 - 4*a^12 + 7/6*a^10)*s)*t^4 + ((-1/8*a^34 + 2/3*a^30 + 1/6*a^28 - 23/12*a^26 + 11/6*a^24 - 15/4*a^22 + 9/2*a^20 - 73/24*a^18 + 29/12*a^14 - 11/6*a^12 + 5/12*a^10)*s^2 + (1/4*a^42 + 1/2*a^38 - 6*a^36 - 65/8*a^34 + 60*a^32 - 137/2*a^30 + 11/2*a^28 + 39/2*a^26 - 23/2*a^24 + 73/4*a^22 - 39/2*a^20 + 85/8*a^18 - 29/4*a^14 + 11/2*a^12 - 5/4*a^10)*s)*t^3 + ((1/60*a^50 - 1/2*a^42 - a^38 + 6*a^36 + 35/4*a^34 - 40*a^32 + 94/3*a^30 + 1/3*a^28 - 7/2*a^26 + 8/3*a^24 - 20/3*a^22 + 7*a^20 - 9/2*a^18 + 3*a^14 - 2*a^12 + 2/5*a^10)*s)*t^2 + ((-1/60*a^50 + 1/4*a^42 + 1/2*a^38 - 2*a^36 - 3*a^34 + 10*a^32 - 6*a^30)*s)*t)*w^5 + ((-1/6*a^22*s^3 + (a^28 - 2*a^26 + 2/3*a^22)*s^2 + (-5*a^28 + 5*a^26 - 5/6*a^22)*s)*t^5 + (1/6*a^22*s^3 + (a^30 - 2*a^28 - a^26 + 4*a^24 - 3/2*a^22)*s^2 + (-4*a^30 + 12*a^28 + 2*a^26 - 8*a^24 + 8/3*a^22)*s)*t^4 + ((1/6*a^34 - a^30 + 2*a^26 - a^24 + 5/6*a^22)*s^2 + (-1/2*a^34 + 6*a^30 - 6*a^28 - 9*a^26 + 3*a^24 - 5/2*a^22)*s)*t^3 + ((-1/3*a^28 + 3*a^24 - 4*a^22)*s^2 + (1/3*a^34 - 2*a^30 + 2/3*a^28 + 3*a^26 - 4*a^24 + 26/3*a^22)*s)*t^2 + ((-1/3*a^28 + 3*a^24 - 4*a^22)*s)*t)*w^4 + ((1/2*a^22*s^3 - 3/2*a^22*s^2 + 2*a^22*s)*t^4 + (-a^18*s^3 + (1/2*a^26 - a^24 + 3/2*a^22 + 3*a^18)*s^2 + (-3/2*a^26 + 3*a^24 - 9/2*a^22 - 3*a^18)*s)*t^3 + ((-a^22 + 2*a^20 - 3*a^18)*s^2 + (a^26 - 2*a^24 + 4*a^22 - 4*a^20 + 6*a^18)*s)*t^2 + ((-a^22 + 2*a^20 - 2*a^18)*s)*t)*w^3]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Coefficients of the CGF of labeled digraphs,\n",
    "# with the marking variable t for strongly connected components\n",
    "# and the marking variable s for source-like components (we take a parameter a = sqrt{2})\n",
    "QDTSzero = [qdts[i] * a^(i*(i-1)) for i in srange(disp)]\n",
    "QDTSzero"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[s*t,\n",
       " 4*s*t^2*w - 4*s*t*w,\n",
       " 2*s^2*t^3*w^2 + 18*s*t^3*w^2 + 4*s^2*t^2*w - 28*s*t^2*w^2 - 8*s*t^2*w + 8*s*t*w^2 + 4*s*t*w,\n",
       " 4/3*s^3*t^4*w^3 + 28*s^2*t^4*w^3 - 28*s^2*t^3*w^3 + 400/3*s*t^4*w^3 + 32*s^2*t^3*w^2 - 204*s*t^3*w^3 - 32*s^2*t^2*w^2 - 96*s*t^3*w^2 + 112*s*t^2*w^3 + 128*s*t^2*w^2 - 128/3*s*t*w^3 - 32*s*t*w^2,\n",
       " 2/3*s^4*t^5*w^4 + 40*s^3*t^5*w^4 - 116/3*s^3*t^4*w^4 + 516*s^2*t^5*w^4 - 652*s^2*t^4*w^4 + 1810*s*t^5*w^4 + 256*s^2*t^4*w^3 + 178*s^2*t^3*w^4 - 6608/3*s*t^4*w^4 + 32*s^3*t^3*w^2 - 256*s^2*t^3*w^3 - 1536*s*t^4*w^3 + 1962*s*t^3*w^4 - 96*s^2*t^3*w^2 + 128*s^2*t^2*w^3 + 2304*s*t^3*w^3 - 248*s*t^2*w^4 + 96*s^2*t^2*w^2 + 96*s*t^3*w^2 - 1024*s*t^2*w^3 - 4096/3*s*t*w^4 - 192*s*t^2*w^2 + 128*s*t*w^3 + 64*s*t*w^2,\n",
       " 4/15*s^5*t^6*w^5 + 124/3*s^4*t^6*w^5 - 40*s^4*t^5*w^5 + 4616/3*s^3*t^6*w^5 - 5920/3*s^3*t^5*w^5 + 47864/3*s^2*t^6*w^5 - 1024/3*s^3*t^5*w^4 + 1436/3*s^3*t^4*w^5 - 18416*s^2*t^5*w^5 + 234248/5*s*t^6*w^5 + 1024/3*s^3*t^4*w^4 + 4096/3*s^2*t^5*w^4 + 8932*s^2*t^4*w^5 - 108200/3*s*t^5*w^5 + 1024*s^3*t^4*w^3 + 5120*s^2*t^4*w^4 - 128000/3*s*t^5*w^4 - 4424*s^2*t^3*w^5 + 200816/3*s*t^4*w^5 - 512*s^3*t^3*w^3 - 3072*s^2*t^4*w^3 + 3072*s^2*t^3*w^4 + 163840/3*s*t^4*w^4 + 61144*s*t^3*w^5 + 4608*s^2*t^3*w^3 + 4096*s*t^4*w^3 - 4096/3*s^2*t^2*w^4 - 33792*s*t^3*w^4 + 271808/3*s*t^2*w^5 - 1536*s^2*t^2*w^3 - 10752*s*t^3*w^3 + 45056/3*s*t^2*w^4 - 3473408/15*s*t*w^5 + 7168*s*t^2*w^3 - 4096/3*s*t*w^4 - 1024*s*t*w^3]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Coefficients of the CGF of labeled digraphs\n",
    "# with the marking variable t for strongly connected components\n",
    "# and the marking variable s for source-like components\n",
    "QDTSzerosub = [sum(sum(sum(QDTSzero[i][k][j][l].subs(a=sqrt(2)) * s^l * t^j * w^k for l in srange(N)) for j in srange(N)) for k in srange(N)) for i in srange(disp)]\n",
    "QDTSzerosub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[s*t, 0, 0, 0, 0, 0],\n",
       " [0, 4*s*t^2 - 4*s*t, 0, 0, 0, 0],\n",
       " [0, 4*(s^2 - 2*s)*t^2 + 4*s*t, 2*(s^2 + 9*s)*t^3 - 28*s*t^2 + 8*s*t, 0, 0, 0],\n",
       " [0,\n",
       "  0,\n",
       "  32*(s^2 - 3*s)*t^3 - 32*(s^2 - 4*s)*t^2 - 32*s*t,\n",
       "  4/3*(s^3 + 21*s^2 + 100*s)*t^4 - 4*(7*s^2 + 51*s)*t^3 + 112*s*t^2 - 128/3*s*t,\n",
       "  0,\n",
       "  0],\n",
       " [0,\n",
       "  0,\n",
       "  32*(s^3 - 3*s^2 + 3*s)*t^3 + 96*(s^2 - 2*s)*t^2 + 64*s*t,\n",
       "  256*(s^2 - 6*s)*t^4 - 256*(s^2 - 9*s)*t^3 + 128*(s^2 - 8*s)*t^2 + 128*s*t,\n",
       "  2/3*(s^4 + 60*s^3 + 774*s^2 + 2715*s)*t^5 - 4/3*(29*s^3 + 489*s^2 + 1652*s)*t^4 + 2*(89*s^2 + 981*s)*t^3 - 248*s*t^2 - 4096/3*s*t,\n",
       "  0],\n",
       " [0,\n",
       "  0,\n",
       "  0,\n",
       "  1024*(s^3 - 3*s^2 + 4*s)*t^4 - 512*(s^3 - 9*s^2 + 21*s)*t^3 - 512*(3*s^2 - 14*s)*t^2 - 1024*s*t,\n",
       "  -1024/3*(s^3 - 4*s^2 + 125*s)*t^5 + 1024/3*(s^3 + 15*s^2 + 160*s)*t^4 + 3072*(s^2 - 11*s)*t^3 - 4096/3*(s^2 - 11*s)*t^2 - 4096/3*s*t,\n",
       "  4/15*(s^5 + 155*s^4 + 5770*s^3 + 59830*s^2 + 175686*s)*t^6 - 8/3*(15*s^4 + 740*s^3 + 6906*s^2 + 13525*s)*t^5 + 4/3*(359*s^3 + 6699*s^2 + 50204*s)*t^4 - 8*(553*s^2 - 7643*s)*t^3 + 271808/3*s*t^2 - 3473408/15*s*t]]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Coefficients of the CGF of labeled digraphs in matrix form (Section A.9)\n",
    "# the marking variable t is for strongly connected components\n",
    "# and the marking variable s is for source-like components\n",
    "QDTSzeroMatrix = [[QDTSzero[i][j].subs(a=sqrt(2)) for j in srange(disp)] for i in srange(disp)]\n",
    "QDTSzeroMatrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[t, 0, 0, 0, 0, 0],\n",
       " [0, 4*t^2 - 4*t, 0, 0, 0, 0],\n",
       " [0, -4*t^2 + 4*t, 20*t^3 - 28*t^2 + 8*t, 0, 0, 0],\n",
       " [0,\n",
       "  0,\n",
       "  -64*t^3 + 96*t^2 - 32*t,\n",
       "  488/3*t^4 - 232*t^3 + 112*t^2 - 128/3*t,\n",
       "  0,\n",
       "  0],\n",
       " [0,\n",
       "  0,\n",
       "  32*t^3 - 96*t^2 + 64*t,\n",
       "  -1280*t^4 + 2048*t^3 - 896*t^2 + 128*t,\n",
       "  7100/3*t^5 - 8680/3*t^4 + 2140*t^3 - 248*t^2 - 4096/3*t,\n",
       "  0],\n",
       " [0,\n",
       "  0,\n",
       "  0,\n",
       "  2048*t^4 - 6656*t^3 + 5632*t^2 - 1024*t,\n",
       "  -124928/3*t^5 + 180224/3*t^4 - 30720*t^3 + 40960/3*t^2 - 4096/3*t,\n",
       "  965768/15*t^6 - 56496*t^5 + 229048/3*t^4 + 56720*t^3 + 271808/3*t^2 - 3473408/15*t]]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Coefficients of the CGF of labeled digraphs in matrix form (Table 9)\n",
    "# the marking variable t is for strongly connected components\n",
    "QDTzeroMatrix = [[QDTSzeroMatrix[i][j].subs(s=1) for j in srange(disp)] for i in srange(disp)]\n",
    "QDTzeroMatrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[s, 0, 0, 0, 0, 0],\n",
       " [0, 0, 0, 0, 0, 0],\n",
       " [0, 4*s^2 - 4*s, 2*s^2 - 2*s, 0, 0, 0],\n",
       " [0, 0, 0, 4/3*s^3 - 4/3*s, 0, 0],\n",
       " [0, 0, 32*s^3 - 32*s, 128*s^2 - 128*s, 2/3*s^4 + 4/3*s^3 + 42*s^2 - 44*s, 0],\n",
       " [0,\n",
       "  0,\n",
       "  0,\n",
       "  512*s^3 - 512*s,\n",
       "  8192*s^2 - 8192*s,\n",
       "  4/15*s^5 + 4/3*s^4 + 44*s^3 + 6140/3*s^2 - 31384/15*s]]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Coefficients of the CGF of labeled digraphs in matrix form (Table 12)\n",
    "# the marking variable s is for source-like components\n",
    "QDSzeroMatrix = [[QDTSzeroMatrix[i][j].subs(t=1) for j in srange(disp)] for i in srange(disp)]\n",
    "QDSzeroMatrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[s, 0, 0, 0, 0, 0],\n",
       " [0, -4*s, 0, 0, 0, 0],\n",
       " [0, 4*s, 8*s, 0, 0, 0],\n",
       " [0, 0, -32*s, -128/3*s, 0, 0],\n",
       " [0, 0, 64*s, 128*s, -4096/3*s, 0],\n",
       " [0, 0, 0, -1024*s, -4096/3*s, -3473408/15*s]]"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Coefficients of the CGF of labeled digraphs with one strongly connected component (Table 4)\n",
    "# the marking variable s is for source-like components\n",
    "QDTSzeroMatrixPrimT = [[QDTSzeroMatrix[i][j].diff(t).subs(t=0) for j in srange(disp)] for i in srange(disp)]\n",
    "QDTSzeroMatrixPrimT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[0, 0, 0, 0, 0, 0],\n",
       " [0, 4*s, 0, 0, 0, 0],\n",
       " [0, 4*s^2 - 8*s, -28*s, 0, 0, 0],\n",
       " [0, 0, -32*s^2 + 128*s, 112*s, 0, 0],\n",
       " [0, 0, 96*s^2 - 192*s, 128*s^2 - 1024*s, -248*s, 0],\n",
       " [0, 0, 0, -1536*s^2 + 7168*s, -4096/3*s^2 + 45056/3*s, 271808/3*s]]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Coefficients of the CGF of labeled digraphs with two strongly connected components (Table 13)\n",
    "# the marking variable s is for source-like components\n",
    "QDTSzeroMatrixSecT = [[QDTSzeroMatrix[i][j].diff(t,2).subs(t=0) / 2 for j in srange(disp)] for i in srange(disp)]\n",
    "QDTSzeroMatrixSecT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[0, 0, 0, 0, 0, 0],\n",
       " [0, 0, 0, 0, 0, 0],\n",
       " [0, 0, 2*s^2 + 18*s, 0, 0, 0],\n",
       " [0, 0, 32*s^2 - 96*s, -28*s^2 - 204*s, 0, 0],\n",
       " [0, 0, 32*s^3 - 96*s^2 + 96*s, -256*s^2 + 2304*s, 178*s^2 + 1962*s, 0],\n",
       " [0,\n",
       "  0,\n",
       "  0,\n",
       "  -512*s^3 + 4608*s^2 - 10752*s,\n",
       "  3072*s^2 - 33792*s,\n",
       "  -4424*s^2 + 61144*s]]"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Coefficients of the CGF of labeled digraphs with three strongly connected components (Table 14)\n",
    "# the marking variable s is for source-like components\n",
    "QDTSzeroMatrixThirT = [[QDTSzeroMatrix[i][j].diff(t,3).subs(t=0) / 6 for j in srange(disp)] for i in srange(disp)]\n",
    "QDTSzeroMatrixThirT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[t, 0, 0, 0, 0, 0],\n",
       " [0, 4*t^2 - 4*t, 0, 0, 0, 0],\n",
       " [0, -8*t^2 + 4*t, 18*t^3 - 28*t^2 + 8*t, 0, 0, 0],\n",
       " [0,\n",
       "  0,\n",
       "  -96*t^3 + 128*t^2 - 32*t,\n",
       "  400/3*t^4 - 204*t^3 + 112*t^2 - 128/3*t,\n",
       "  0,\n",
       "  0],\n",
       " [0,\n",
       "  0,\n",
       "  96*t^3 - 192*t^2 + 64*t,\n",
       "  -1536*t^4 + 2304*t^3 - 1024*t^2 + 128*t,\n",
       "  1810*t^5 - 6608/3*t^4 + 1962*t^3 - 248*t^2 - 4096/3*t,\n",
       "  0],\n",
       " [0,\n",
       "  0,\n",
       "  0,\n",
       "  4096*t^4 - 10752*t^3 + 7168*t^2 - 1024*t,\n",
       "  -128000/3*t^5 + 163840/3*t^4 - 33792*t^3 + 45056/3*t^2 - 4096/3*t,\n",
       "  234248/5*t^6 - 108200/3*t^5 + 200816/3*t^4 + 61144*t^3 + 271808/3*t^2 - 3473408/15*t]]"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Coefficients of the CGF of labeled digraphs with one strongly connected source-like component (Table 15)\n",
    "# the marking variable t is for strongly connected components\n",
    "QDTSzeroMatrixPrimS = [[QDTSzeroMatrix[i][j].diff(s).subs(s=0) for j in srange(disp)] for i in srange(disp)]\n",
    "QDTSzeroMatrixPrimS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[0, 0, 0, 0, 0, 0],\n",
       " [0, 0, 0, 0, 0, 0],\n",
       " [0, 4*t^2, 2*t^3, 0, 0, 0],\n",
       " [0, 0, 32*t^3 - 32*t^2, 28*t^4 - 28*t^3, 0, 0],\n",
       " [0,\n",
       "  0,\n",
       "  -96*t^3 + 96*t^2,\n",
       "  256*t^4 - 256*t^3 + 128*t^2,\n",
       "  516*t^5 - 652*t^4 + 178*t^3,\n",
       "  0],\n",
       " [0,\n",
       "  0,\n",
       "  0,\n",
       "  -3072*t^4 + 4608*t^3 - 1536*t^2,\n",
       "  4096/3*t^5 + 5120*t^4 + 3072*t^3 - 4096/3*t^2,\n",
       "  47864/3*t^6 - 18416*t^5 + 8932*t^4 - 4424*t^3]]"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Coefficients of the CGF of labeled digraphs with two strongly connected source-like components (Table 16)\n",
    "# the marking variable t is for strongly connected components\n",
    "QDTSzeroMatrixSecS = [[QDTSzeroMatrix[i][j].diff(s,2).subs(s=0) / 2 for j in srange(disp)] for i in srange(disp)]\n",
    "QDTSzeroMatrixSecS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[0, 0, 0, 0, 0, 0],\n",
       " [0, 0, 0, 0, 0, 0],\n",
       " [0, 0, 0, 0, 0, 0],\n",
       " [0, 0, 0, 4/3*t^4, 0, 0],\n",
       " [0, 0, 32*t^3, 0, 40*t^5 - 116/3*t^4, 0],\n",
       " [0,\n",
       "  0,\n",
       "  0,\n",
       "  1024*t^4 - 512*t^3,\n",
       "  -1024/3*t^5 + 1024/3*t^4,\n",
       "  4616/3*t^6 - 5920/3*t^5 + 1436/3*t^4]]"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Coefficients of the CGF of labeled digraphs with three strongly connected source-like components (Table 17)\n",
    "# the marking variable t is for strongly connected components\n",
    "QDTSzeroMatrixThirS = [[QDTSzeroMatrix[i][j].diff(s,3).subs(s=0) / 6 for j in srange(disp)] for i in srange(disp)]\n",
    "QDTSzeroMatrixThirS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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