stirlingReps.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Code for Stirling reps of the first and second kinds\n",
    "\n",
    "**WARNING: THE FUNCTIONS FOR THE ORLIK-SOLOMON ALGEBRA MAY NOT WORK IF YOU ARE NOT USING SAGEMATH 10.0 OR LATER**\n",
    "\n",
    "Run the first cell with NO CHANGES.\n",
    "\n",
    "Later cells contain demos for obtaining families of representations for OS and VG of the Type A braid arrangement.\n",
    "\n",
    "Functions include:\n",
    "\n",
    "1. `koszulDualReps(primalReps, q)` : input a list of symmetric group characters corresponding to graded pieces of an algebra $A$. Outputs a list of characters for the Koszul dual up to degree $q$.\n",
    "2. `stirlingFirstVG(n,k)`\n",
    "3. `stirlingFirstOS(n,k)`\n",
    "4. `stirlingSecondVG(n,k)`\n",
    "5. `stirlingSecondOS(n,k)`\n",
    "6. `rankTwoRepsOS(n)` : Characters for the graded pieces of $OS(U_{2,n})$\n",
    "7. `rankTwoDualRepsOS(n,q)`: Characters for the graded pieces of $OS(U_{2,n})^!$ up to degree $q$\n",
    "8. `equivariantDeviationsSkew(L)` : input a list of characters for the graded pieces of A^! up to degree n WHERE A IS **SKEW-COMMUTATIVE**. Outputs list of chracters for primitives / indecomposables.\n",
    "9. `equivariantDeviationsCommutative(L)` input a list of characters for the graded pieces of A^! up to degree n WHERE A IS **COMMUTATIVE**. Outputs list of characters for primitives / indecomposable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "########################\n",
    "### DECLARATIONS ###\n",
    "########################\n",
    "\n",
    "Sym = SymmetricFunctions(QQ);\n",
    "\n",
    "# need to pick a basis to do anything meaningful\n",
    "s = Sym.s() # Schur basis\n",
    "h = Sym.h() # homogeneous basis\n",
    "m = Sym.m() # monomial basis\n",
    "p = Sym.p() # power sum basis\n",
    "e = Sym.e() # elementary basis\n",
    "\n",
    "\n",
    "# declare basis w.r.t. homogeneous polynomials for defining H\n",
    "#R.<t,z> = PowerSeriesRing(SymmetricFunctions(QQ).h())\n",
    "\n",
    "############################################################################\n",
    "\n",
    "\n",
    "##########################################\n",
    "### KOSZUL DUAL REPRESENTATIONS ###\n",
    "##########################################\n",
    "\n",
    "#INPUT: A list of representations L, and a number p up to which we will compute the duals\n",
    "#OUTPUT: The representations for the Koszul dual [A_i]^!\n",
    "#CAVEAT: The output only makes sense and is only guaranteed to be schur-positive if A is Koszul\n",
    "\n",
    "def koszulDualReps(primalReps,q):\n",
    "    returnedReps = [];\n",
    "    n = len(primalReps);\n",
    "    for i in range(q): \n",
    "        if i == 0: \n",
    "            returnedReps.append(s([n]))\n",
    "            continue \n",
    "        if i == 1: \n",
    "            returnedReps.append(primalReps[1])\n",
    "            continue\n",
    "        totalRep = 0\n",
    "        for j in range(1,min(i,n)): \n",
    "            sgn = (-1)**(j+1)\n",
    "            totalRep += sgn * primalReps[j].itensor(returnedReps[i-j])\n",
    "        if (i < n):\n",
    "            totalRep += (-1)**(i+1) * primalReps[i]\n",
    "        returnedReps.append(totalRep)\n",
    "    return returnedReps\n",
    "\n",
    "\n",
    "##########################################\n",
    "### STIRLING REPRESENTATIONS ###\n",
    "##########################################\n",
    "\n",
    "#INPUT: integers n,k\n",
    "#OUTPUT: The Stirling rep c_{VG}(n,k) of the first kind\n",
    "def stirlingFirstVG(n,k):\n",
    "    L = Partitions(n,length=n-k).list();\n",
    "    return sum(s.gessel_reutenauer(x) for x in L);\n",
    "\n",
    "#INPUT: integers n,k\n",
    "#OUTPUT: The Stirling rep c_{OS}(n,k) of the first kind\n",
    "def stirlingFirstOS(n,k):   \n",
    "    L = Partitions(n,length=n-k).list();\n",
    "    return sum(s.lehrer_solomon(x) for x in L)\n",
    "\n",
    "#INPUT: integers n,p\n",
    "#OUTPUT: A list of Stirling reps S_{OS}(n,i) of the second kind from 0 to p\n",
    "#q = how far out to compute to\n",
    "def stirlingSecondOS(n,q):\n",
    "    returnedReps = [];\n",
    "    for i in range(q): \n",
    "        if i == 0: \n",
    "            returnedReps.append(s([n]))\n",
    "            continue \n",
    "        primalReps = [stirlingFirstOS(n,k) for k in range(0,n)];\n",
    "        return koszulDualReps(primalReps,q)\n",
    "\n",
    "#INPUT: integers n,p\n",
    "#OUTPUT: A list of Stirling reps S_{VG}(n,i) of the second kind from 0 to p\n",
    "#q = how far out to compute to\n",
    "def stirlingSecondVG(n,q):\n",
    "    returnedReps = [];\n",
    "    for i in range(q): \n",
    "        if i == 0: \n",
    "            returnedReps.append(s([n]))\n",
    "            continue \n",
    "        primalReps = [stirlingFirstVG(n,k) for k in range(0,n)];\n",
    "        return koszulDualReps(primalReps,q)\n",
    "\n",
    "############################\n",
    "### RANK TWO MATROIDS ###\n",
    "############################\n",
    "\n",
    "#INPUT: n the size of the ground set\n",
    "#OUTPUT: Representations for the graded pieces of OS(U_{2,n})\n",
    "def rankTwoRepsOS(n):\n",
    "    return [s[n],s[n]+s[n-1,1],s[n-1,1]]\n",
    "\n",
    "#INPUT: n the size of the ground set\n",
    "#OUTPUT: Representations for the graded pieces of OS(U_{2,n})\n",
    "def rankTwoDualRepsOS(n,q):\n",
    "    return koszulDualReps(rankTwoRepsOS(n),q)\n",
    "\n",
    "\n",
    "#################################\n",
    "### EQUIVARIANT DEVIATIONS ###\n",
    "#################################\n",
    "\n",
    "#INPUT: A list L of representations for the graded pieces of A^! up to degree n WHERE A IS **SKEW-COMMUTATIVE**\n",
    "#OUTPUT: The equivariant deviations up to degree n\n",
    "def equivariantDeviationsSkew(L):\n",
    "    n = len(L);\n",
    "    returnedReps = [0,L[1]];\n",
    "    for i in range(2,n): \n",
    "        totalRep = L[i];\n",
    "        for q in Partitions(i, min_length=2):\n",
    "            expPart = q.to_exp()\n",
    "            tensorList = [];\n",
    "            for j in range(0,len(expPart)):\n",
    "                tensorList.append(s[expPart[j]].inner_plethysm(returnedReps[j+1]));\n",
    "            repTensor = tensorList[0];\n",
    "            if (len(tensorList)>1):\n",
    "                for j in range (1,len(tensorList)):\n",
    "                    repTensor = repTensor.itensor(tensorList[j]);\n",
    "            totalRep += -repTensor\n",
    "        returnedReps.append(totalRep)\n",
    "    return returnedReps\n",
    "\n",
    "#INPUT: A list L of representations for the graded pieces of A^! up to degree n WHERE A IS **COMMUTATIVE**\n",
    "#OUTPUT: The equivariant deviations up to degree n\n",
    "def equivariantDeviationsCommutative(L):\n",
    "    n = len(L);\n",
    "    returnedReps = [0,L[1]];\n",
    "    for i in range(2,n): \n",
    "        totalRep = L[i];\n",
    "        for q in Partitions(i, min_length=2):\n",
    "            expPart = q.to_exp()\n",
    "            tensorList = [];\n",
    "            for j in range(0,len(expPart)):\n",
    "                if (j%2==1):\n",
    "                    tensorList.append(s[expPart[j]].inner_plethysm(returnedReps[j+1]));\n",
    "                if (j%2==0):\n",
    "                    tensorList.append(e[expPart[j]].inner_plethysm(returnedReps[j+1]));\n",
    "            repTensor = tensorList[0];\n",
    "            if (len(tensorList)>1):\n",
    "                for j in range (1,len(tensorList)):\n",
    "                    repTensor = repTensor.itensor(tensorList[j]);\n",
    "            totalRep += -repTensor\n",
    "        returnedReps.append(totalRep)\n",
    "    return returnedReps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "equivariant deviations for j = 2:\n",
      " n = 2: 0\n",
      " n = 3: s[1, 1, 1]\n",
      " n = 4: s[2, 1, 1]\n",
      " n = 5: s[3, 1, 1]\n",
      "equivariant deviations for j = 3:\n",
      " n = 2: 0\n",
      " n = 3: s[2, 1]\n",
      " n = 4: s[2, 1, 1] + s[2, 2] + s[3, 1]\n",
      " n = 5: s[2, 2, 1] + s[3, 1, 1] + s[3, 2] + s[4, 1]\n",
      "equivariant deviations for j = 4:\n",
      " n = 2: 0\n",
      " n = 3: s[1, 1, 1] + s[2, 1]\n",
      " n = 4: s[1, 1, 1, 1] + 3*s[2, 1, 1] + s[2, 2] + 2*s[3, 1]\n",
      " n = 5: 2*s[2, 1, 1, 1] + 2*s[2, 2, 1] + 4*s[3, 1, 1] + 2*s[3, 2] + 2*s[4, 1]\n",
      "equivariant deviations for j = 5:\n",
      " n = 2: 0\n",
      " n = 3: s[1, 1, 1] + 2*s[2, 1] + s[3]\n",
      " n = 4: 2*s[1, 1, 1, 1] + 6*s[2, 1, 1] + 4*s[2, 2] + 6*s[3, 1] + 2*s[4]\n",
      " n = 5: s[1, 1, 1, 1, 1] + 6*s[2, 1, 1, 1] + 8*s[2, 2, 1] + 10*s[3, 1, 1] + 9*s[3, 2] + 8*s[4, 1] + 2*s[5]\n"
     ]
    }
   ],
   "source": [
    "#loop for displaying rank 2 matroid equivariant deviations\n",
    "a = 6\n",
    "n = 6\n",
    "eqDevs = [];\n",
    "for i in range(2,n): \n",
    "    eqDevs.append(equivariantDeviationsSkew(rankTwoDualRepsOS(i,a)));\n",
    "\n",
    "for j in range(2,a):\n",
    "    print(\"equivariant deviations for j = \" + str(j) + \":\")\n",
    "    for k in range(len(eqDevs)):\n",
    "              print(\" n = \" + str(k+2) + \": \" + str((eqDevs[k][j])))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "equivariant deviations for j = 1:\n",
      " n = 2: s[1, 1]\n",
      " n = 3: s[1, 1, 1] + s[2, 1]\n",
      " n = 4: s[2, 1, 1] + s[3, 1]\n",
      " n = 5: s[3, 1, 1] + s[4, 1]\n",
      " n = 6: s[4, 1, 1] + s[5, 1]\n",
      " n = 7: s[5, 1, 1] + s[6, 1]\n",
      "equivariant deviations for j = 2:\n",
      " n = 2: s[2]\n",
      " n = 3: s[2, 1] + 2*s[3]\n",
      " n = 4: s[2, 2] + 2*s[3, 1] + 2*s[4]\n",
      " n = 5: 2*s[3, 2] + 2*s[4, 1] + 2*s[5]\n",
      " n = 6: s[3, 3] + 2*s[4, 2] + 2*s[5, 1] + 2*s[6]\n",
      " n = 7: s[4, 3] + 2*s[5, 2] + 2*s[6, 1] + 2*s[7]\n",
      "equivariant deviations for j = 3:\n",
      " n = 2: 0\n",
      " n = 3: s[2, 1]\n",
      " n = 4: s[2, 1, 1] + 2*s[2, 2] + s[3, 1]\n",
      " n = 5: 2*s[2, 2, 1] + s[3, 1, 1] + 2*s[3, 2] + s[4, 1]\n",
      " n = 6: s[2, 2, 2] + 2*s[3, 2, 1] + s[4, 1, 1] + 2*s[4, 2] + s[5, 1]\n",
      " n = 7: s[3, 2, 2] + 2*s[4, 2, 1] + s[5, 1, 1] + 2*s[5, 2] + s[6, 1]\n",
      "equivariant deviations for j = 4:\n",
      " n = 2: 0\n",
      " n = 3: s[1, 1, 1] + s[2, 1]\n",
      " n = 4: s[1, 1, 1, 1] + 4*s[2, 1, 1] + s[2, 2] + 2*s[3, 1]\n",
      " n = 5: 3*s[2, 1, 1, 1] + 3*s[2, 2, 1] + 6*s[3, 1, 1] + 2*s[3, 2] + 2*s[4, 1]\n",
      " n = 6: 2*s[2, 2, 1, 1] + 4*s[3, 1, 1, 1] + 5*s[3, 2, 1] + s[3, 3] + 6*s[4, 1, 1] + 2*s[4, 2] + 2*s[5, 1]\n",
      " n = 7: 3*s[3, 2, 1, 1] + 2*s[3, 3, 1] + 4*s[4, 1, 1, 1] + 5*s[4, 2, 1] + s[4, 3] + 6*s[5, 1, 1] + 2*s[5, 2] + 2*s[6, 1]\n",
      "equivariant deviations for j = 5:\n",
      " n = 2: 0\n",
      " n = 3: s[1, 1, 1] + 2*s[2, 1] + s[3]\n",
      " n = 4: 2*s[1, 1, 1, 1] + 7*s[2, 1, 1] + 4*s[2, 2] + 7*s[3, 1] + 2*s[4]\n",
      " n = 5: s[1, 1, 1, 1, 1] + 8*s[2, 1, 1, 1] + 10*s[2, 2, 1] + 13*s[3, 1, 1] + 11*s[3, 2] + 10*s[4, 1] + 2*s[5]\n",
      " n = 6: 3*s[2, 1, 1, 1, 1] + 9*s[2, 2, 1, 1] + 4*s[2, 2, 2] + 10*s[3, 1, 1, 1] + 18*s[3, 2, 1] + 8*s[3, 3] + 16*s[4, 1, 1] + 14*s[4, 2] + 10*s[5, 1] + 2*s[6]\n",
      " n = 7: 3*s[2, 2, 1, 1, 1] + 3*s[2, 2, 2, 1] + 3*s[3, 1, 1, 1, 1] + 11*s[3, 2, 1, 1] + 6*s[3, 2, 2] + 9*s[3, 3, 1] + 11*s[4, 1, 1, 1] + 21*s[4, 2, 1] + 11*s[4, 3] + 16*s[5, 1, 1] + 14*s[5, 2] + 10*s[6, 1] + 2*s[7]\n",
      "equivariant deviations for j = 6:\n",
      " n = 2: 0\n",
      " n = 3: s[1, 1, 1] + 4*s[2, 1] + 2*s[3]\n",
      " n = 4: 4*s[1, 1, 1, 1] + 15*s[2, 1, 1] + 14*s[2, 2] + 17*s[3, 1] + 7*s[4]\n",
      " n = 5: 6*s[1, 1, 1, 1, 1] + 22*s[2, 1, 1, 1] + 36*s[2, 2, 1] + 37*s[3, 1, 1] + 39*s[3, 2] + 31*s[4, 1] + 10*s[5]\n",
      " n = 6: 3*s[1, 1, 1, 1, 1, 1] + 17*s[2, 1, 1, 1, 1] + 33*s[2, 2, 1, 1] + 25*s[2, 2, 2] + 37*s[3, 1, 1, 1] + 77*s[3, 2, 1] + 26*s[3, 3] + 49*s[4, 1, 1] + 58*s[4, 2] + 36*s[5, 1] + 11*s[6]\n",
      " n = 7: 6*s[2, 1, 1, 1, 1, 1] + 14*s[2, 2, 1, 1, 1] + 18*s[2, 2, 2, 1] + 22*s[3, 1, 1, 1, 1] + 56*s[3, 2, 1, 1] + 45*s[3, 2, 2] + 42*s[3, 3, 1] + 40*s[4, 1, 1, 1] + 95*s[4, 2, 1] + 45*s[4, 3] + 51*s[5, 1, 1] + 64*s[5, 2] + 37*s[6, 1] + 11*s[7]\n"
     ]
    }
   ],
   "source": [
    "#loop for displaying VG equivariant deviations\n",
    "n = 8\n",
    "a = 7\n",
    "eqDevs = [];\n",
    "for i in range(2,n): \n",
    "    eqDevs.append(equivariantDeviationsCommutative(stirlingSecondVG(i,a)));\n",
    "\n",
    "for j in range(1,a):\n",
    "    print(\"equivariant deviations for j = \" + str(j) + \":\")\n",
    "    for k in range(len(eqDevs)):\n",
    "              print(\" n = \" + str(k+2) + \": \" + str((eqDevs[k][j])))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "ename": "AttributeError",
     "evalue": "'SymmetricFunctionAlgebra_schur_with_category' object has no attribute 'lehrer_solomon'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[0;32m/usr/lib/python3/dist-packages/sage/structure/category_object.pyx\u001b[0m in \u001b[0;36msage.structure.category_object.CategoryObject.getattr_from_category (build/cythonized/sage/structure/category_object.c:7158)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    838\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 839\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__cached_methods\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    840\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyError\u001b[0m: 'lehrer_solomon'",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m/tmp/ipykernel_41210/2654718588.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[0meqDevs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      5\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 6\u001b[0;31m     \u001b[0meqDevs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mequivariantDeviationsSkew\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mstirlingSecondOS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      7\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      8\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/tmp/ipykernel_41210/3116073170.py\u001b[0m in \u001b[0;36mstirlingSecondOS\u001b[0;34m(n, q)\u001b[0m\n\u001b[1;32m     72\u001b[0m             \u001b[0mreturnedReps\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     73\u001b[0m             \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 74\u001b[0;31m         \u001b[0mprimalReps\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mstirlingFirstOS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     75\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mkoszulDualReps\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mprimalReps\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     76\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/tmp/ipykernel_41210/3116073170.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m     72\u001b[0m             \u001b[0mreturnedReps\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     73\u001b[0m             \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 74\u001b[0;31m         \u001b[0mprimalReps\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mstirlingFirstOS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     75\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mkoszulDualReps\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mprimalReps\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     76\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/tmp/ipykernel_41210/3116073170.py\u001b[0m in \u001b[0;36mstirlingFirstOS\u001b[0;34m(n, k)\u001b[0m\n\u001b[1;32m     61\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mstirlingFirstOS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     62\u001b[0m     \u001b[0mL\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPartitions\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mlength\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 63\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0msum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlehrer_solomon\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mx\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mL\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     64\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     65\u001b[0m \u001b[0;31m#INPUT: integers n,p\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/lib/python3/dist-packages/sage/misc/functional.py\u001b[0m in \u001b[0;36msymbolic_sum\u001b[0;34m(expression, *args, **kwds)\u001b[0m\n\u001b[1;32m    574\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mexpression\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    575\u001b[0m     \u001b[0;32melif\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m<=\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 576\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0msum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpression\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    577\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    578\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msymbolic\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mring\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mSR\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/tmp/ipykernel_41210/3116073170.py\u001b[0m in \u001b[0;36m<genexpr>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m     61\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mstirlingFirstOS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     62\u001b[0m     \u001b[0mL\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPartitions\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mlength\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 63\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0msum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlehrer_solomon\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mx\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mL\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     64\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     65\u001b[0m \u001b[0;31m#INPUT: integers n,p\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/lib/python3/dist-packages/sage/structure/category_object.pyx\u001b[0m in \u001b[0;36msage.structure.category_object.CategoryObject.__getattr__ (build/cythonized/sage/structure/category_object.c:7077)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    831\u001b[0m             \u001b[0mAttributeError\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0;34m'PrimeNumbers_with_category'\u001b[0m \u001b[0mobject\u001b[0m \u001b[0mhas\u001b[0m \u001b[0mno\u001b[0m \u001b[0mattribute\u001b[0m \u001b[0;34m'sadfasdf'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    832\u001b[0m         \"\"\"\n\u001b[0;32m--> 833\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgetattr_from_category\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    834\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    835\u001b[0m     \u001b[0mcdef\u001b[0m \u001b[0mgetattr_from_category\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/lib/python3/dist-packages/sage/structure/category_object.pyx\u001b[0m in \u001b[0;36msage.structure.category_object.CategoryObject.getattr_from_category (build/cythonized/sage/structure/category_object.c:7243)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    846\u001b[0m                 \u001b[0mcls\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_category\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mparent_class\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    847\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 848\u001b[0;31m             \u001b[0mattr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgetattr_from_other_class\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    849\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__cached_methods\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mattr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    850\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mattr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/lib/python3/dist-packages/sage/cpython/getattr.pyx\u001b[0m in \u001b[0;36msage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2636)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    354\u001b[0m         \u001b[0mdummy_error_message\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcls\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    355\u001b[0m         \u001b[0mdummy_error_message\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mname\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 356\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mAttributeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdummy_error_message\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    357\u001b[0m     \u001b[0mcdef\u001b[0m \u001b[0mPyObject\u001b[0m\u001b[0;34m*\u001b[0m \u001b[0mattr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0minstance_getattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    358\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mattr\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mNULL\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mAttributeError\u001b[0m: 'SymmetricFunctionAlgebra_schur_with_category' object has no attribute 'lehrer_solomon'"
     ]
    }
   ],
   "source": [
    "#loop for displaying OS equivariant deviations\n",
    "#NOTE: This function may not work for you if you are not using SageMath 10.0 or higher\n",
    "n = 6\n",
    "a = 6\n",
    "eqDevs = [];\n",
    "for i in range(2,n): \n",
    "    eqDevs.append(equivariantDeviationsSkew(stirlingSecondOS(i,a)));\n",
    "\n",
    "for j in range(2,a):\n",
    "    print(\"equivariant deviations for j = \" + str(j) + \":\")\n",
    "    for k in range(len(eqDevs)):\n",
    "              print(\" n = \" + str(k+2) + \": \" + str(eqDevs[k][j]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "representations of VG(A_0):\n",
      "    degree 0: s[1]\n",
      "    degree 1: 0\n",
      "    degree 2: 0\n",
      "    degree 3: 0\n",
      "    degree 4: 0\n",
      "    degree 5: 0\n",
      "representations of VG(A_1):\n",
      "    degree 0: s[2]\n",
      "    degree 1: s[1, 1]\n",
      "    degree 2: 0\n",
      "    degree 3: 0\n",
      "    degree 4: 0\n",
      "    degree 5: 0\n",
      "representations of VG(A_2):\n",
      "    degree 0: s[3]\n",
      "    degree 1: s[1, 1, 1] + s[2, 1]\n",
      "    degree 2: s[2, 1]\n",
      "    degree 3: 0\n",
      "    degree 4: 0\n",
      "    degree 5: 0\n",
      "representations of VG(A_3):\n",
      "    degree 0: s[4]\n",
      "    degree 1: s[2, 1, 1] + s[3, 1]\n",
      "    degree 2: s[1, 1, 1, 1] + s[2, 1, 1] + 2*s[2, 2] + s[3, 1]\n",
      "    degree 3: s[2, 1, 1] + s[3, 1]\n",
      "    degree 4: 0\n",
      "    degree 5: 0\n",
      "representations of VG(A_4):\n",
      "    degree 0: s[5]\n",
      "    degree 1: s[3, 1, 1] + s[4, 1]\n",
      "    degree 2: s[1, 1, 1, 1, 1] + s[2, 1, 1, 1] + 2*s[2, 2, 1] + s[3, 1, 1] + 2*s[3, 2] + s[4, 1]\n",
      "    degree 3: 2*s[2, 1, 1, 1] + 2*s[2, 2, 1] + 3*s[3, 1, 1] + 2*s[3, 2] + s[4, 1]\n",
      "    degree 4: s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] + s[4, 1]\n",
      "    degree 5: 0\n",
      "representations of VG(A_5):\n",
      "    degree 0: s[6]\n",
      "    degree 1: s[4, 1, 1] + s[5, 1]\n",
      "    degree 2: s[2, 1, 1, 1, 1] + s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[4, 1, 1] + 2*s[4, 2] + s[5, 1]\n",
      "    degree 3: s[1, 1, 1, 1, 1, 1] + s[2, 1, 1, 1, 1] + 4*s[2, 2, 1, 1] + s[2, 2, 2] + 3*s[3, 1, 1, 1] + 5*s[3, 2, 1] + 3*s[3, 3] + 3*s[4, 1, 1] + 2*s[4, 2] + s[5, 1]\n",
      "    degree 4: 2*s[2, 1, 1, 1, 1] + 3*s[2, 2, 1, 1] + 3*s[2, 2, 2] + 5*s[3, 1, 1, 1] + 6*s[3, 2, 1] + s[3, 3] + 3*s[4, 1, 1] + 4*s[4, 2] + s[5, 1]\n",
      "    degree 5: s[2, 1, 1, 1, 1] + 2*s[2, 2, 1, 1] + s[3, 1, 1, 1] + 3*s[3, 2, 1] + s[3, 3] + 2*s[4, 1, 1] + s[4, 2] + s[5, 1]\n",
      "representations of VG(A_6):\n",
      "    degree 0: s[7]\n",
      "    degree 1: s[5, 1, 1] + s[6, 1]\n",
      "    degree 2: s[3, 1, 1, 1, 1] + s[3, 2, 2] + s[4, 1, 1, 1] + 2*s[4, 2, 1] + s[5, 1, 1] + 2*s[5, 2] + s[6, 1]\n",
      "    degree 3: s[1, 1, 1, 1, 1, 1, 1] + s[2, 1, 1, 1, 1, 1] + 2*s[2, 2, 1, 1, 1] + 2*s[2, 2, 2, 1] + s[3, 1, 1, 1, 1] + 5*s[3, 2, 1, 1] + 2*s[3, 2, 2] + 4*s[3, 3, 1] + 3*s[4, 1, 1, 1] + 5*s[4, 2, 1] + 3*s[4, 3] + 3*s[5, 1, 1] + 2*s[5, 2] + s[6, 1]\n",
      "    degree 4: 2*s[2, 1, 1, 1, 1, 1] + 4*s[2, 2, 1, 1, 1] + 5*s[2, 2, 2, 1] + 6*s[3, 1, 1, 1, 1] + 12*s[3, 2, 1, 1] + 8*s[3, 2, 2] + 6*s[3, 3, 1] + 6*s[4, 1, 1, 1] + 11*s[4, 2, 1] + 5*s[4, 3] + 3*s[5, 1, 1] + 4*s[5, 2] + s[6, 1]\n",
      "    degree 5: 2*s[2, 1, 1, 1, 1, 1] + 6*s[2, 2, 1, 1, 1] + 5*s[2, 2, 2, 1] + 5*s[3, 1, 1, 1, 1] + 13*s[3, 2, 1, 1] + 7*s[3, 2, 2] + 8*s[3, 3, 1] + 7*s[4, 1, 1, 1] + 12*s[4, 2, 1] + 4*s[4, 3] + 5*s[5, 1, 1] + 4*s[5, 2] + s[6, 1]\n"
     ]
    }
   ],
   "source": [
    "#loop for displaying VG reps\n",
    "n = 7\n",
    "l = 6\n",
    "for i in range(1, n+1): \n",
    "    print(\"representations of VG(A_\" + str(i-1) + \"):\")\n",
    "    reps = [];\n",
    "    for j in range(0,l) :\n",
    "      reps.append(stirlingFirstVG(i,j));\n",
    "    for k in range(len(reps)): \n",
    "        print(\"    degree \" + str(k) + \": \" + str(reps[k]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "representations of OS(A_1)^!:\n"
     ]
    },
    {
     "ename": "AttributeError",
     "evalue": "'SymmetricFunctionAlgebra_schur_with_category' object has no attribute 'lehrer_solomon'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[0;32m/usr/lib/python3/dist-packages/sage/structure/category_object.pyx\u001b[0m in \u001b[0;36msage.structure.category_object.CategoryObject.getattr_from_category (build/cythonized/sage/structure/category_object.c:7158)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    838\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 839\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__cached_methods\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    840\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyError\u001b[0m: 'lehrer_solomon'",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m/tmp/ipykernel_41210/1330515463.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      5\u001b[0m     \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"representations of OS(A_\"\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m\")^!:\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 6\u001b[0;31m     \u001b[0mreps\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mstirlingSecondOS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ml\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      7\u001b[0m     \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mreps\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      8\u001b[0m         \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"    degree \"\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m\": \"\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mreps\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/tmp/ipykernel_41210/3116073170.py\u001b[0m in \u001b[0;36mstirlingSecondOS\u001b[0;34m(n, q)\u001b[0m\n\u001b[1;32m     72\u001b[0m             \u001b[0mreturnedReps\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     73\u001b[0m             \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 74\u001b[0;31m         \u001b[0mprimalReps\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mstirlingFirstOS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     75\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mkoszulDualReps\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mprimalReps\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     76\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/tmp/ipykernel_41210/3116073170.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m     72\u001b[0m             \u001b[0mreturnedReps\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     73\u001b[0m             \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 74\u001b[0;31m         \u001b[0mprimalReps\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mstirlingFirstOS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     75\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mkoszulDualReps\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mprimalReps\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     76\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/tmp/ipykernel_41210/3116073170.py\u001b[0m in \u001b[0;36mstirlingFirstOS\u001b[0;34m(n, k)\u001b[0m\n\u001b[1;32m     61\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mstirlingFirstOS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     62\u001b[0m     \u001b[0mL\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPartitions\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mlength\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 63\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0msum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlehrer_solomon\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mx\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mL\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     64\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     65\u001b[0m \u001b[0;31m#INPUT: integers n,p\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/lib/python3/dist-packages/sage/misc/functional.py\u001b[0m in \u001b[0;36msymbolic_sum\u001b[0;34m(expression, *args, **kwds)\u001b[0m\n\u001b[1;32m    574\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mexpression\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    575\u001b[0m     \u001b[0;32melif\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m<=\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 576\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0msum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpression\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    577\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    578\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msymbolic\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mring\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mSR\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/tmp/ipykernel_41210/3116073170.py\u001b[0m in \u001b[0;36m<genexpr>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m     61\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mstirlingFirstOS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     62\u001b[0m     \u001b[0mL\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPartitions\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mlength\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 63\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0msum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlehrer_solomon\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mx\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mL\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     64\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     65\u001b[0m \u001b[0;31m#INPUT: integers n,p\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/lib/python3/dist-packages/sage/structure/category_object.pyx\u001b[0m in \u001b[0;36msage.structure.category_object.CategoryObject.__getattr__ (build/cythonized/sage/structure/category_object.c:7077)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    831\u001b[0m             \u001b[0mAttributeError\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0;34m'PrimeNumbers_with_category'\u001b[0m \u001b[0mobject\u001b[0m \u001b[0mhas\u001b[0m \u001b[0mno\u001b[0m \u001b[0mattribute\u001b[0m \u001b[0;34m'sadfasdf'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    832\u001b[0m         \"\"\"\n\u001b[0;32m--> 833\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgetattr_from_category\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    834\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    835\u001b[0m     \u001b[0mcdef\u001b[0m \u001b[0mgetattr_from_category\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/lib/python3/dist-packages/sage/structure/category_object.pyx\u001b[0m in \u001b[0;36msage.structure.category_object.CategoryObject.getattr_from_category (build/cythonized/sage/structure/category_object.c:7243)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    846\u001b[0m                 \u001b[0mcls\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_category\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mparent_class\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    847\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 848\u001b[0;31m             \u001b[0mattr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgetattr_from_other_class\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    849\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__cached_methods\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mattr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    850\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mattr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/lib/python3/dist-packages/sage/cpython/getattr.pyx\u001b[0m in \u001b[0;36msage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2636)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    354\u001b[0m         \u001b[0mdummy_error_message\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcls\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    355\u001b[0m         \u001b[0mdummy_error_message\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mname\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 356\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mAttributeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdummy_error_message\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    357\u001b[0m     \u001b[0mcdef\u001b[0m \u001b[0mPyObject\u001b[0m\u001b[0;34m*\u001b[0m \u001b[0mattr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0minstance_getattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    358\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mattr\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mNULL\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mAttributeError\u001b[0m: 'SymmetricFunctionAlgebra_schur_with_category' object has no attribute 'lehrer_solomon'"
     ]
    }
   ],
   "source": [
    "#Lists OS(A_n)^! for first n\n",
    "#NOTE: This may not work for you if you are not using SageMath 10.0 or higher\n",
    "n = 7\n",
    "l = 8\n",
    "for j in range(2, n+1): \n",
    "    print(\"representations of OS(A_\" + str(j-1) + \")^!:\")\n",
    "    reps = stirlingSecondOS(j,l);\n",
    "    for k in range(len(reps)): \n",
    "        print(\"    degree \" + str(k+1) + \": \" + str(reps[k]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "representations of VG(A_1)^!:\n",
      "    degree 0: s[2]\n",
      "    degree 1: s[1, 1]\n",
      "    degree 2: s[2]\n",
      "    degree 3: s[1, 1]\n",
      "    degree 4: s[2]\n",
      "    degree 5: s[1, 1]\n",
      "    degree 6: s[2]\n",
      "    degree 7: s[1, 1]\n",
      "    degree 8: s[2]\n",
      "representations of VG(A_2)^!:\n",
      "    degree 0: s[3]\n",
      "    degree 1: s[1, 1, 1] + s[2, 1]\n",
      "    degree 2: s[1, 1, 1] + 2*s[2, 1] + 2*s[3]\n",
      "    degree 3: 3*s[1, 1, 1] + 5*s[2, 1] + 2*s[3]\n",
      "    degree 4: 5*s[1, 1, 1] + 10*s[2, 1] + 6*s[3]\n",
      "    degree 5: 11*s[1, 1, 1] + 21*s[2, 1] + 10*s[3]\n",
      "    degree 6: 21*s[1, 1, 1] + 42*s[2, 1] + 22*s[3]\n",
      "    degree 7: 43*s[1, 1, 1] + 85*s[2, 1] + 42*s[3]\n",
      "    degree 8: 85*s[1, 1, 1] + 170*s[2, 1] + 86*s[3]\n",
      "representations of VG(A_3)^!:\n",
      "    degree 0: s[4]\n",
      "    degree 1: s[2, 1, 1] + s[3, 1]\n",
      "    degree 2: s[1, 1, 1, 1] + 3*s[2, 1, 1] + 2*s[2, 2] + 3*s[3, 1] + 2*s[4]\n",
      "    degree 3: 4*s[1, 1, 1, 1] + 11*s[2, 1, 1] + 8*s[2, 2] + 11*s[3, 1] + 4*s[4]\n",
      "    degree 4: 12*s[1, 1, 1, 1] + 38*s[2, 1, 1] + 24*s[2, 2] + 38*s[3, 1] + 13*s[4]\n",
      "    degree 5: 40*s[1, 1, 1, 1] + 121*s[2, 1, 1] + 80*s[2, 2] + 121*s[3, 1] + 40*s[4]\n",
      "    degree 6: 127*s[1, 1, 1, 1] + 377*s[2, 1, 1] + 254*s[2, 2] + 377*s[3, 1] + 128*s[4]\n",
      "    degree 7: 388*s[1, 1, 1, 1] + 1167*s[2, 1, 1] + 776*s[2, 2] + 1167*s[3, 1] + 388*s[4]\n",
      "    degree 8: 1186*s[1, 1, 1, 1] + 3564*s[2, 1, 1] + 2372*s[2, 2] + 3564*s[3, 1] + 1187*s[4]\n",
      "representations of VG(A_4)^!:\n",
      "    degree 0: s[5]\n",
      "    degree 1: s[3, 1, 1] + s[4, 1]\n",
      "    degree 2: 2*s[2, 1, 1, 1] + 2*s[2, 2, 1] + 3*s[3, 1, 1] + 3*s[3, 2] + 3*s[4, 1] + 2*s[5]\n",
      "    degree 3: 3*s[1, 1, 1, 1, 1] + 10*s[2, 1, 1, 1] + 14*s[2, 2, 1] + 17*s[3, 1, 1] + 15*s[3, 2] + 14*s[4, 1] + 4*s[5]\n",
      "    degree 4: 12*s[1, 1, 1, 1, 1] + 54*s[2, 1, 1, 1] + 68*s[2, 2, 1] + 86*s[3, 1, 1] + 72*s[3, 2] + 60*s[4, 1] + 17*s[5]\n",
      "    degree 5: 62*s[1, 1, 1, 1, 1] + 252*s[2, 1, 1, 1] + 320*s[2, 2, 1] + 389*s[3, 1, 1] + 326*s[3, 2] + 267*s[4, 1] + 68*s[5]\n",
      "    degree 6: 278*s[1, 1, 1, 1, 1] + 1124*s[2, 1, 1, 1] + 1414*s[2, 2, 1] + 1703*s[3, 1, 1] + 1429*s[3, 2] + 1151*s[4, 1] + 294*s[5]\n",
      "    degree 7: 1201*s[1, 1, 1, 1, 1] + 4830*s[2, 1, 1, 1] + 6058*s[2, 2, 1] + 7289*s[3, 1, 1] + 6085*s[3, 2] + 4888*s[4, 1] + 1228*s[5]\n",
      "    degree 8: 5066*s[1, 1, 1, 1, 1] + 20328*s[2, 1, 1, 1] + 25448*s[2, 2, 1] + 30578*s[3, 1, 1] + 25506*s[3, 2] + 20440*s[4, 1] + 5125*s[5]\n",
      "representations of VG(A_5)^!:\n",
      "    degree 0: s[6]\n",
      "    degree 1: s[4, 1, 1] + s[5, 1]\n",
      "    degree 2: s[2, 2, 1, 1] + 2*s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[3, 3] + 3*s[4, 1, 1] + 3*s[4, 2] + 3*s[5, 1] + 2*s[6]\n",
      "    degree 3: s[1, 1, 1, 1, 1, 1] + 5*s[2, 1, 1, 1, 1] + 8*s[2, 2, 1, 1] + 7*s[2, 2, 2] + 13*s[3, 1, 1, 1] + 21*s[3, 2, 1] + 7*s[3, 3] + 18*s[4, 1, 1] + 18*s[4, 2] + 14*s[5, 1] + 4*s[6]\n",
      "    degree 4: 7*s[1, 1, 1, 1, 1, 1] + 37*s[2, 1, 1, 1, 1] + 74*s[2, 2, 1, 1] + 40*s[2, 2, 2] + 89*s[3, 1, 1, 1] + 150*s[3, 2, 1] + 52*s[3, 3] + 109*s[4, 1, 1] + 100*s[4, 2] + 67*s[5, 1] + 18*s[6]\n",
      "    degree 5: 49*s[1, 1, 1, 1, 1, 1] + 257*s[2, 1, 1, 1, 1] + 488*s[2, 2, 1, 1] + 275*s[2, 2, 2] + 563*s[3, 1, 1, 1] + 935*s[3, 2, 1] + 309*s[3, 3] + 629*s[4, 1, 1] + 574*s[4, 2] + 351*s[5, 1] + 78*s[6]\n",
      "    degree 6: 302*s[1, 1, 1, 1, 1, 1] + 1590*s[2, 1, 1, 1, 1] + 2935*s[2, 2, 1, 1] + 1665*s[2, 2, 2] + 3346*s[3, 1, 1, 1] + 5463*s[3, 2, 1] + 1746*s[3, 3] + 3526*s[4, 1, 1] + 3235*s[4, 2] + 1872*s[5, 1] + 405*s[6]\n",
      "    degree 7: 1788*s[1, 1, 1, 1, 1, 1] + 9176*s[2, 1, 1, 1, 1] + 16800*s[2, 2, 1, 1] + 9420*s[2, 2, 2] + 18886*s[3, 1, 1, 1] + 30614*s[3, 2, 1] + 9706*s[3, 3] + 19470*s[4, 1, 1] + 17684*s[4, 2] + 10056*s[5, 1] + 2082*s[6]\n",
      "    degree 8: 10031*s[1, 1, 1, 1, 1, 1] + 50861*s[2, 1, 1, 1, 1] + 92548*s[2, 2, 1, 1] + 51628*s[2, 2, 2] + 103411*s[3, 1, 1, 1] + 166776*s[3, 2, 1] + 52588*s[3, 3] + 105241*s[4, 1, 1] + 95164*s[4, 2] + 53567*s[5, 1] + 10906*s[6]\n",
      "representations of VG(A_6)^!:\n",
      "    degree 0: s[7]\n",
      "    degree 1: s[5, 1, 1] + s[6, 1]\n",
      "    degree 2: s[3, 2, 1, 1] + 2*s[4, 1, 1, 1] + 2*s[4, 2, 1] + s[4, 3] + 3*s[5, 1, 1] + 3*s[5, 2] + 3*s[6, 1] + 2*s[7]\n",
      "    degree 3: s[2, 1, 1, 1, 1, 1] + 2*s[2, 2, 1, 1, 1] + 3*s[2, 2, 2, 1] + 6*s[3, 1, 1, 1, 1] + 11*s[3, 2, 1, 1] + 8*s[3, 2, 2] + 7*s[3, 3, 1] + 13*s[4, 1, 1, 1] + 22*s[4, 2, 1] + 10*s[4, 3] + 18*s[5, 1, 1] + 18*s[5, 2] + 14*s[6, 1] + 4*s[7]\n",
      "    degree 4: 2*s[1, 1, 1, 1, 1, 1, 1] + 15*s[2, 1, 1, 1, 1, 1] + 38*s[2, 2, 1, 1, 1] + 39*s[2, 2, 2, 1] + 49*s[3, 1, 1, 1, 1] + 127*s[3, 2, 1, 1] + 76*s[3, 2, 2] + 87*s[3, 3, 1] + 99*s[4, 1, 1, 1] + 181*s[4, 2, 1] + 80*s[4, 3] + 113*s[5, 1, 1] + 107*s[5, 2] + 68*s[6, 1] + 18*s[7]\n",
      "    degree 5: 22*s[1, 1, 1, 1, 1, 1, 1] + 146*s[2, 1, 1, 1, 1, 1] + 371*s[2, 2, 1, 1, 1] + 386*s[2, 2, 2, 1] + 440*s[3, 1, 1, 1, 1] + 1102*s[3, 2, 1, 1] + 679*s[3, 2, 2] + 730*s[3, 3, 1] + 741*s[4, 1, 1, 1] + 1359*s[4, 2, 1] + 585*s[4, 3] + 717*s[5, 1, 1] + 685*s[5, 2] + 372*s[6, 1] + 80*s[7]\n",
      "    degree 6: 186*s[1, 1, 1, 1, 1, 1, 1] + 1241*s[2, 1, 1, 1, 1, 1] + 3073*s[2, 2, 1, 1, 1] + 3184*s[2, 2, 2, 1] + 3500*s[3, 1, 1, 1, 1] + 8571*s[3, 2, 1, 1] + 5275*s[3, 2, 2] + 5478*s[3, 3, 1] + 5344*s[4, 1, 1, 1] + 9717*s[4, 2, 1] + 4082*s[4, 3] + 4655*s[5, 1, 1] + 4470*s[5, 2] + 2193*s[6, 1] + 438*s[7]\n",
      "    degree 7: 1490*s[1, 1, 1, 1, 1, 1, 1] + 9539*s[2, 1, 1, 1, 1, 1] + 23231*s[2, 2, 1, 1, 1] + 23815*s[2, 2, 2, 1] + 25704*s[3, 1, 1, 1, 1] + 62139*s[3, 2, 1, 1] + 37933*s[3, 2, 2] + 38883*s[3, 3, 1] + 37272*s[4, 1, 1, 1] + 67029*s[4, 2, 1] + 27693*s[4, 3] + 30644*s[5, 1, 1] + 29135*s[5, 2] + 13529*s[6, 1] + 2502*s[7]\n",
      "    degree 8: 10961*s[1, 1, 1, 1, 1, 1, 1] + 68573*s[2, 1, 1, 1, 1, 1] + 164785*s[2, 2, 1, 1, 1] + 167612*s[2, 2, 2, 1] + 179780*s[3, 1, 1, 1, 1] + 429842*s[3, 2, 1, 1] + 260933*s[3, 2, 2] + 265055*s[3, 3, 1] + 252607*s[4, 1, 1, 1] + 450324*s[4, 2, 1] + 183950*s[4, 3] + 200464*s[5, 1, 1] + 189417*s[5, 2] + 85135*s[6, 1] + 15110*s[7]\n",
      "representations of VG(A_7)^!:\n",
      "    degree 0: s[8]\n",
      "    degree 1: s[6, 1, 1] + s[7, 1]\n",
      "    degree 2: s[4, 2, 1, 1] + 2*s[5, 1, 1, 1] + 2*s[5, 2, 1] + s[5, 3] + 3*s[6, 1, 1] + 3*s[6, 2] + 3*s[7, 1] + 2*s[8]\n",
      "    degree 3: s[2, 2, 2, 2] + s[3, 1, 1, 1, 1, 1] + 3*s[3, 2, 1, 1, 1] + 3*s[3, 2, 2, 1] + 3*s[3, 3, 1, 1] + s[3, 3, 2] + 6*s[4, 1, 1, 1, 1] + 11*s[4, 2, 1, 1] + 8*s[4, 2, 2] + 8*s[4, 3, 1] + 3*s[4, 4] + 13*s[5, 1, 1, 1] + 22*s[5, 2, 1] + 10*s[5, 3] + 18*s[6, 1, 1] + 18*s[6, 2] + 14*s[7, 1] + 4*s[8]\n",
      "    degree 4: 3*s[2, 1, 1, 1, 1, 1, 1] + 10*s[2, 2, 1, 1, 1, 1] + 15*s[2, 2, 2, 1, 1] + 7*s[2, 2, 2, 2] + 17*s[3, 1, 1, 1, 1, 1] + 54*s[3, 2, 1, 1, 1] + 60*s[3, 2, 2, 1] + 54*s[3, 3, 1, 1] + 41*s[3, 3, 2] + 52*s[4, 1, 1, 1, 1] + 139*s[4, 2, 1, 1] + 84*s[4, 2, 2] + 118*s[4, 3, 1] + 28*s[4, 4] + 100*s[5, 1, 1, 1] + 185*s[5, 2, 1] + 87*s[5, 3] + 113*s[6, 1, 1] + 108*s[6, 2] + 68*s[7, 1] + 18*s[8]\n",
      "    degree 5: 5*s[1, 1, 1, 1, 1, 1, 1, 1] + 50*s[2, 1, 1, 1, 1, 1, 1] + 155*s[2, 2, 1, 1, 1, 1] + 235*s[2, 2, 2, 1, 1] + 115*s[2, 2, 2, 2] + 206*s[3, 1, 1, 1, 1, 1] + 681*s[3, 2, 1, 1, 1] + 771*s[3, 2, 2, 1] + 681*s[3, 3, 1, 1] + 531*s[3, 3, 2] + 506*s[4, 1, 1, 1, 1] + 1389*s[4, 2, 1, 1] + 875*s[4, 2, 2] + 1195*s[4, 3, 1] + 276*s[4, 4] + 782*s[5, 1, 1, 1] + 1485*s[5, 2, 1] + 699*s[5, 3] + 732*s[6, 1, 1] + 707*s[6, 2] + 375*s[7, 1] + 80*s[8]\n",
      "    degree 6: 71*s[1, 1, 1, 1, 1, 1, 1, 1] + 581*s[2, 1, 1, 1, 1, 1, 1] + 1836*s[2, 2, 1, 1, 1, 1] + 2691*s[2, 2, 2, 1, 1] + 1389*s[2, 2, 2, 2] + 2165*s[3, 1, 1, 1, 1, 1] + 7135*s[3, 2, 1, 1, 1] + 8116*s[3, 2, 2, 1] + 6975*s[3, 3, 1, 1] + 5370*s[3, 3, 2] + 4667*s[4, 1, 1, 1, 1] + 12701*s[4, 2, 1, 1] + 8137*s[4, 2, 2] + 10760*s[4, 3, 1] + 2424*s[4, 4] + 6188*s[5, 1, 1, 1] + 11794*s[5, 2, 1] + 5480*s[5, 3] + 4997*s[6, 1, 1] + 4905*s[6, 2] + 2264*s[7, 1] + 443*s[8]\n",
      "    degree 7: 759*s[1, 1, 1, 1, 1, 1, 1, 1] + 5965*s[2, 1, 1, 1, 1, 1, 1] + 18364*s[2, 2, 1, 1, 1, 1] + 26842*s[2, 2, 2, 1, 1] + 13719*s[2, 2, 2, 2] + 20655*s[3, 1, 1, 1, 1, 1] + 67021*s[3, 2, 1, 1, 1] + 75858*s[3, 2, 2, 1] + 63588*s[3, 3, 1, 1] + 48872*s[3, 3, 2] + 40689*s[4, 1, 1, 1, 1] + 109954*s[4, 2, 1, 1] + 70069*s[4, 2, 2] + 91239*s[4, 3, 1] + 19792*s[4, 4] + 49006*s[5, 1, 1, 1] + 92976*s[5, 2, 1] + 42666*s[5, 3] + 35913*s[6, 1, 1] + 35052*s[6, 2] + 14761*s[7, 1] + 2614*s[8]\n",
      "    degree 8: 7272*s[1, 1, 1, 1, 1, 1, 1, 1] + 55397*s[2, 1, 1, 1, 1, 1, 1] + 167283*s[2, 2, 1, 1, 1, 1] + 242590*s[2, 2, 2, 1, 1] + 123221*s[2, 2, 2, 2] + 183177*s[3, 1, 1, 1, 1, 1] + 585555*s[3, 2, 1, 1, 1] + 658086*s[3, 2, 2, 1] + 542845*s[3, 3, 1, 1] + 415242*s[3, 3, 2] + 341059*s[4, 1, 1, 1, 1] + 912015*s[4, 2, 1, 1] + 577980*s[4, 2, 2] + 743521*s[4, 3, 1] + 157134*s[4, 4] + 385750*s[5, 1, 1, 1] + 726376*s[5, 2, 1] + 329320*s[5, 3] + 264435*s[6, 1, 1] + 256665*s[6, 2] + 101451*s[7, 1] + 16760*s[8]\n",
      "representations of VG(A_8)^!:\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    degree 0: s[9]\n",
      "    degree 1: s[7, 1, 1] + s[8, 1]\n",
      "    degree 2: s[5, 2, 1, 1] + 2*s[6, 1, 1, 1] + 2*s[6, 2, 1] + s[6, 3] + 3*s[7, 1, 1] + 3*s[7, 2] + 3*s[8, 1] + 2*s[9]\n",
      "    degree 3: s[3, 2, 2, 2] + s[3, 3, 1, 1, 1] + s[4, 1, 1, 1, 1, 1] + 3*s[4, 2, 1, 1, 1] + 3*s[4, 2, 2, 1] + 3*s[4, 3, 1, 1] + s[4, 3, 2] + s[4, 4, 1] + 6*s[5, 1, 1, 1, 1] + 11*s[5, 2, 1, 1] + 8*s[5, 2, 2] + 8*s[5, 3, 1] + 3*s[5, 4] + 13*s[6, 1, 1, 1] + 22*s[6, 2, 1] + 10*s[6, 3] + 18*s[7, 1, 1] + 18*s[7, 2] + 14*s[8, 1] + 4*s[9]\n",
      "    degree 4: s[2, 2, 1, 1, 1, 1, 1] + 2*s[2, 2, 2, 1, 1, 1] + 2*s[2, 2, 2, 2, 1] + 3*s[3, 1, 1, 1, 1, 1, 1] + 12*s[3, 2, 1, 1, 1, 1] + 20*s[3, 2, 2, 1, 1] + 10*s[3, 2, 2, 2] + 16*s[3, 3, 1, 1, 1] + 22*s[3, 3, 2, 1] + 5*s[3, 3, 3] + 18*s[4, 1, 1, 1, 1, 1] + 57*s[4, 2, 1, 1, 1] + 62*s[4, 2, 2, 1] + 66*s[4, 3, 1, 1] + 49*s[4, 3, 2] + 31*s[4, 4, 1] + 52*s[5, 1, 1, 1, 1] + 140*s[5, 2, 1, 1] + 84*s[5, 2, 2] + 122*s[5, 3, 1] + 35*s[5, 4] + 100*s[6, 1, 1, 1] + 185*s[6, 2, 1] + 88*s[6, 3] + 113*s[7, 1, 1] + 108*s[7, 2] + 68*s[8, 1] + 18*s[9]\n",
      "    degree 5: 9*s[2, 1, 1, 1, 1, 1, 1, 1] + 36*s[2, 2, 1, 1, 1, 1, 1] + 73*s[2, 2, 2, 1, 1, 1] + 65*s[2, 2, 2, 2, 1] + 61*s[3, 1, 1, 1, 1, 1, 1] + 242*s[3, 2, 1, 1, 1, 1] + 391*s[3, 2, 2, 1, 1] + 200*s[3, 2, 2, 2] + 327*s[3, 3, 1, 1, 1] + 467*s[3, 3, 2, 1] + 128*s[3, 3, 3] + 220*s[4, 1, 1, 1, 1, 1] + 778*s[4, 2, 1, 1, 1] + 896*s[4, 2, 2, 1] + 983*s[4, 3, 1, 1] + 769*s[4, 3, 2] + 465*s[4, 4, 1] + 517*s[5, 1, 1, 1, 1] + 1443*s[5, 2, 1, 1] + 914*s[5, 2, 2] + 1324*s[5, 3, 1] + 390*s[5, 4] + 787*s[6, 1, 1, 1] + 1501*s[6, 2, 1] + 721*s[6, 3] + 733*s[7, 1, 1] + 710*s[7, 2] + 375*s[8, 1] + 80*s[9]\n",
      "    degree 6: 17*s[1, 1, 1, 1, 1, 1, 1, 1, 1] + 169*s[2, 1, 1, 1, 1, 1, 1, 1] + 662*s[2, 2, 1, 1, 1, 1, 1] + 1256*s[2, 2, 2, 1, 1, 1] + 1131*s[2, 2, 2, 2, 1] + 845*s[3, 1, 1, 1, 1, 1, 1] + 3516*s[3, 2, 1, 1, 1, 1] + 5670*s[3, 2, 2, 1, 1] + 3006*s[3, 2, 2, 2] + 4675*s[3, 3, 1, 1, 1] + 6769*s[3, 3, 2, 1] + 1797*s[3, 3, 3] + 2560*s[4, 1, 1, 1, 1, 1] + 9237*s[4, 2, 1, 1, 1] + 10885*s[4, 2, 2, 1] + 11782*s[4, 3, 1, 1] + 9383*s[4, 3, 2] + 5419*s[4, 4, 1] + 5008*s[5, 1, 1, 1, 1] + 14130*s[5, 2, 1, 1] + 9168*s[5, 2, 2] + 13091*s[5, 3, 1] + 3836*s[5, 4] + 6362*s[6, 1, 1, 1] + 12306*s[6, 2, 1] + 5933*s[6, 3] + 5050*s[7, 1, 1] + 4986*s[7, 2] + 2272*s[8, 1] + 444*s[9]\n",
      "    degree 7: 246*s[1, 1, 1, 1, 1, 1, 1, 1, 1] + 2375*s[2, 1, 1, 1, 1, 1, 1, 1] + 8948*s[2, 2, 1, 1, 1, 1, 1] + 16900*s[2, 2, 2, 1, 1, 1] + 15255*s[2, 2, 2, 2, 1] + 10535*s[3, 1, 1, 1, 1, 1, 1] + 43136*s[3, 2, 1, 1, 1, 1] + 69800*s[3, 2, 2, 1, 1] + 36988*s[3, 2, 2, 2] + 55663*s[3, 3, 1, 1, 1] + 80991*s[3, 3, 2, 1] + 21441*s[3, 3, 3] + 27839*s[4, 1, 1, 1, 1, 1] + 100523*s[4, 2, 1, 1, 1] + 119006*s[4, 2, 2, 1] + 126472*s[4, 3, 1, 1] + 101005*s[4, 3, 2] + 56026*s[4, 4, 1] + 47490*s[5, 1, 1, 1, 1] + 134820*s[5, 2, 1, 1] + 87613*s[5, 2, 2] + 124216*s[5, 3, 1] + 35556*s[5, 4] + 52948*s[6, 1, 1, 1] + 102860*s[6, 2, 1] + 49464*s[6, 3] + 37269*s[7, 1, 1] + 36759*s[7, 2] + 15008*s[8, 1] + 2630*s[9]\n",
      "    degree 8: 3115*s[1, 1, 1, 1, 1, 1, 1, 1, 1] + 28636*s[2, 1, 1, 1, 1, 1, 1, 1] + 105312*s[2, 2, 1, 1, 1, 1, 1] + 197433*s[2, 2, 2, 1, 1, 1] + 177755*s[2, 2, 2, 2, 1] + 118280*s[3, 1, 1, 1, 1, 1, 1] + 476951*s[3, 2, 1, 1, 1, 1] + 768446*s[3, 2, 2, 1, 1] + 406449*s[3, 2, 2, 2] + 599102*s[3, 3, 1, 1, 1] + 868620*s[3, 3, 2, 1] + 227637*s[3, 3, 3] + 286782*s[4, 1, 1, 1, 1, 1] + 1027455*s[4, 2, 1, 1, 1] + 1213254*s[4, 2, 2, 1] + 1269259*s[4, 3, 1, 1] + 1011113*s[4, 3, 2] + 544524*s[4, 4, 1] + 445237*s[5, 1, 1, 1, 1] + 1259505*s[5, 2, 1, 1] + 817132*s[5, 2, 2] + 1147117*s[5, 3, 1] + 321147*s[5, 4] + 450783*s[6, 1, 1, 1] + 873967*s[6, 2, 1] + 417421*s[6, 3] + 288725*s[7, 1, 1] + 284301*s[7, 2] + 106295*s[8, 1] + 17139*s[9]\n"
     ]
    }
   ],
   "source": [
    "#Lists VG(A_n)^! for first n\n",
    "n = 9\n",
    "for j in range(2, n+1): \n",
    "    print(\"representations of VG(A_\" + str(j-1) + \")^!:\")\n",
    "    reps = stirlingSecondVG(j,n);\n",
    "    for k in range(len(reps)): \n",
    "        print(\"    degree \" + str(k) + \": \" + str(reps[k]))"
   ]
  }
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