{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "### Computing with Short Step Quadrant Models (including Examples 4.3, 4.5, and 4.6)\n", "Code to generate the groups, orbit sums, and diagonal expressions for the short step quadrant models with finite group. \n", "*Requirements: None*" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "# Enter the short step sets defining quadrant models with finite group\n", "N = (0,1); SS = (0,-1); E = (1,0); W = (-1,0);\n", "NE = (1,1); NW = (-1,1); SE = (1,-1); SW = (-1,-1);\n", "HS1 = [N,SS,E,W]\n", "HS2 = [NE,SE,NW,SW]\n", "HS3 = [N,SS,NE,SE,NW,SW]\n", "HS4 = [N,SS,E,W,NW,SW,SE,NE]\n", "PD1 = [NE,NW,SS]\n", "PD2 = [N,NW,NE,SS]\n", "PD3 = [N,NE,NW,SE,SW]\n", "PD4 = [NE,NW,E,W,SS]\n", "PD5 = [N,NW,NE,E,W,SS]\n", "PD6 = [N,E,W,NE,NW,SE,SW]\n", "ND1 = [N,SE,SW]\n", "ND2 = [N,SS,SE,SW]\n", "ND3 = [NE,NW,SE,SW,SS]\n", "ND4 = [N,E,W,SE,SW]\n", "ND5 = [N,E,W,SS,SW,SE]\n", "ND6 = [NE,NW,E,W,SE,SW,SS]\n", "SP1 = [N,W,SE]\n", "SP2 = [NW,SE,N,SS,E,W]\n", "SP3 = [E,SE,W,NW]\n", "ALG1 = [NE,W,SS]\n", "ALG2 = [N,E,SW]\n", "ALG3 = [N,NE,E,SS,SW,W]\n", "ALG4 = [NE,E,SW,W]" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Define functions to plot a step set, generate the group, orbit sum, and diagonal expression for a walk\n", "var('x,y,t,X,Y')\n", "\n", "# Plot a step set S\n", "def plot_steps(S):\n", " pt = sum([arrow((0,0),s) for s in S])\n", " pt.set_aspect_ratio(1)\n", " return pt\n", "\n", "# Generate the group of a walk (up to a fixed number of elements)\n", "def genGP(ST):\n", " # Get the characteristic polynomial of the model\n", " S = sum([X^i*Y^j for [i,j] in ST])\n", "\n", " # Define the rational maps for the lattice path model\n", " Am1 = S.coefficient(Y,-1); A0 = S.coefficient(Y,0); A1 = S.coefficient(Y,1)\n", " Bm1 = S.coefficient(X,-1); B0 = S.coefficient(X,0); B1 = S.coefficient(X,1)\n", " def Ψ(SS): return [L.subs(X=SS[0],Y=SS[1]).simplify() for L in [(1/X)*Bm1/B1,Y]]\n", " def Φ(SS): return [L.subs(X=SS[0],Y=SS[1]).simplify() for L in [X,(1/Y)*Am1/A1]]\n", "\n", " # Keep applying the maps until the set stabilizes or gets larger than a fixed bound\n", " List = [ ([],[x,y]) ]\n", " N = -1\n", " while (N != len(List)) and (len(List) < 10):\n", " N = len(List)\n", " for i in List:\n", " newI = true\n", " newP = true\n", "\n", " t2I = [k.simplify_full() for k in Ψ(i[1])]\n", " t2P = [k.simplify_full() for k in Φ(i[1])]\n", "\n", " for k in List:\n", " if k[1] == t2I:\n", " newI = false\n", " if k[1] == t2P:\n", " newP = false\n", " if not (newI or newP):\n", " break\n", "\n", " if newI:\n", " List = List + [(['Ψ'] + i[0],t2I)]\n", " if newP:\n", " List = List + [(['Φ'] + i[0],t2P)]\n", " if len(List) >= 10:\n", " break\n", "\n", " if len(List) >= 10:\n", " print(\"Warning: Group size is larger than test bound, and may be infinite\")\n", " return List\n", "\n", "# Get orbit sum\n", "def OrbitSum(ST):\n", " K = 1-t*add([x^i*y^j for [i,j] in ST])\n", " LST = genGP(ST)\n", "\n", " # Sum the Kernel Equation after it's acted upon by the group elements\n", " function('F')(Y)\n", " function('G')(X)\n", " function('Q')(X,Y)\n", " var('Kk')\n", "\n", " EQ = Kk*X*Y*Q(X,Y) == X*Y + F(Y) + G(X)\n", " orb_eq = add([EQ.substitute(X=k[0],Y=k[1])*(-1)^len(i) for [i,k] in LST]).simplify()\n", "\n", " # Return the right-hand side after dividing by the kernel\n", " orb_sum = (orb_eq.rhs()/(K*x*y)).factor()\n", " return orb_sum\n", "\n", "# Convert orbit sum into rational diagonal expression\n", "def RatDiag(ST):\n", " return (OrbitSum(ST).subs(x=1/x,y=1/y,t=x*y*t)/(1-x)/(1-y)).factor()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Examples 4.3 and 4.5 (NSEW Walks)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# These examples deal with the step set {N,S,E,W}\n", "ST = (N,SS,E,W)\n", "show(plot_steps(ST),figsize=2, axes=false)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[([], [x, y]), (['Ψ'], [1/x, y]), (['Φ'], [x, 1/y]), (['Φ', 'Ψ'], [1/x, 1/y])]" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# The group of this model consists of the maps taking (x,y) -> (x^(+/-1), y^(+/-1))\n", "genGP(ST)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}x y Q\\left(x, y\\right) - \\frac{x Q\\left(x, \\frac{1}{y}\\right)}{y} - \\frac{y Q\\left(\\frac{1}{x}, y\\right)}{x} + \\frac{Q\\left(\\frac{1}{x}, \\frac{1}{y}\\right)}{x y} = -\\frac{{\\left(x + 1\\right)} {\\left(x - 1\\right)} {\\left(y + 1\\right)} {\\left(y - 1\\right)}}{t x^{2} y + t x y^{2} + t x + t y - x y}\n", "\\end{math}" ], "text/plain": [ "x*y*Q(x, y) - x*Q(x, 1/y)/y - y*Q(1/x, y)/x + Q(1/x, 1/y)/(x*y) == -(x + 1)*(x - 1)*(y + 1)*(y - 1)/(t*x^2*y + t*x*y^2 + t*x + t*y - x*y)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Taking an orbit sum of the kernel equation gives the following identity\n", "K = 1-t*add([x^i*y^j for [i,j] in ST])\n", "LST = genGP(ST)\n", "function('Q')(X,Y)\n", "var('Kk')\n", "EQ = Kk*X*Y*Q(X,Y) == X*Y + t*Y*Q(0,y) + t*X*Q(x,0)\n", "orb_eq = (add([(EQ.substitute(X=k[0],Y=k[1])*(-1)^len(i)/Kk).expand() for [i,k] in LST])).subs(Kk=K)\n", "show(orb_eq.lhs() == orb_eq.rhs().factor())" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|C(t)|\\phantom{\\verb!x!}\\verb|=| \\Delta{\\Big(} -\\frac{{\\left(x + 1\\right)} {\\left(y + 1\\right)}}{t x^{2} y + t x y^{2} + t x + t y - 1} {\\Big)}\n", "\\end{math}" ], "text/plain": [ "'C(t) = ' \\Delta{\\Big(} -(x + 1)*(y + 1)/(t*x^2*y + t*x*y^2 + t*x + t*y - 1) {\\Big)}" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# This gives the stated diagonal expression for the generating function. Using the RatDiag function above,\n", "show(\"C(t) = \", LatexExpr(\"\\\\Delta{\\\\Big(}\"), RatDiag(ST), LatexExpr(\"{\\\\Big)}\"))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example 4.6 (A Zero Orbit Model)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Now we examine the group of Gessel's model\n", "ST = [E,W,SW,NE]\n", "show(plot_steps(ST),figsize=2, axes=false)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[([], [x, y]),\n", " (['Ψ'], [1/(x*y), y]),\n", " (['Φ'], [x, 1/(x^2*y)]),\n", " (['Φ', 'Ψ'], [1/(x*y), x^2*y]),\n", " (['Ψ', 'Φ'], [x*y, 1/(x^2*y)]),\n", " (['Ψ', 'Φ', 'Ψ'], [1/x, x^2*y]),\n", " (['Φ', 'Ψ', 'Φ'], [x*y, 1/y]),\n", " (['Φ', 'Ψ', 'Φ', 'Ψ'], [1/x, 1/y])]" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# As noted, the group contains the element (x,y) -> (x*y, 1/y) of odd order that fixes x*y\n", "genGP(ST)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}x y Q\\left(x, y\\right) - x y Q\\left(\\frac{1}{x}, x^{2} y\\right) - x Q\\left(x y, \\frac{1}{y}\\right) + x Q\\left(\\frac{1}{x y}, x^{2} y\\right) + \\frac{Q\\left(x y, \\frac{1}{x^{2} y}\\right)}{x} - \\frac{Q\\left(\\frac{1}{x y}, y\\right)}{x} - \\frac{Q\\left(x, \\frac{1}{x^{2} y}\\right)}{x y} + \\frac{Q\\left(\\frac{1}{x}, \\frac{1}{y}\\right)}{x y} = 0\n", "\\end{math}" ], "text/plain": [ "x*y*Q(x, y) - x*y*Q(1/x, x^2*y) - x*Q(x*y, 1/y) + x*Q(1/(x*y), x^2*y) + Q(x*y, 1/(x^2*y))/x - Q(1/(x*y), y)/x - Q(x, 1/(x^2*y))/(x*y) + Q(1/x, 1/y)/(x*y) == 0" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# This implies an orbit sum of zero\n", "K = 1-t*add([x^i*y^j for [i,j] in ST])\n", "LST = genGP(ST)\n", "EQ = Kk*X*Y*Q(X,Y) == X*Y + t*Y*Q(0,y) + t*X*Q(x,0)\n", "orb_eq = (add([(EQ.substitute(X=k[0],Y=k[1])*(-1)^len(i)/Kk).expand() for [i,k] in LST])).subs(Kk=K)\n", "show(orb_eq.lhs() == orb_eq.rhs().factor())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Diagonal Expressions for all Short Step Quadrant Models\n", "Separated into the families studied in Chapters 4, 6, and 10" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x + 1\\right)} {\\left(y + 1\\right)}}{t x^{2} y + t x y^{2} + t x + t y - 1}\n", "\\end{math}" ], "text/plain": [ "-(x + 1)*(y + 1)/(t*x^2*y + t*x*y^2 + t*x + t*y - 1)" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x + 1\\right)} {\\left(y + 1\\right)}}{t x^{2} y^{2} + t x^{2} + t y^{2} + t - 1}\n", "\\end{math}" ], "text/plain": [ "-(x + 1)*(y + 1)/(t*x^2*y^2 + t*x^2 + t*y^2 + t - 1)" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 6 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x + 1\\right)} {\\left(y + 1\\right)}}{t x^{2} y^{2} + t x y^{2} + t x^{2} + t y^{2} + t x + t - 1}\n", "\\end{math}" ], "text/plain": [ "-(x + 1)*(y + 1)/(t*x^2*y^2 + t*x*y^2 + t*x^2 + t*y^2 + t*x + t - 1)" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 8 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x + 1\\right)} {\\left(y + 1\\right)}}{t x^{2} y^{2} + t x^{2} y + t x y^{2} + t x^{2} + t y^{2} + t x + t y + t - 1}\n", "\\end{math}" ], "text/plain": [ "-(x + 1)*(y + 1)/(t*x^2*y^2 + t*x^2*y + t*x*y^2 + t*x^2 + t*y^2 + t*x + t*y + t - 1)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Highly symmetric models\n", "for k in [HS1,HS2,HS3,HS4]:\n", " print(\"The generating function for the step set\")\n", " show(plot_steps(k), figsize=2, axes=false)\n", " print(\"is the main power series diagonal of\")\n", " show(RatDiag(k))" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 3 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x y^{2} - x^{2} - 1\\right)} {\\left(x + 1\\right)}}{{\\left(t x y^{2} + t x^{2} + t - 1\\right)} {\\left(x^{2} + 1\\right)} {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x*y^2 - x^2 - 1)*(x + 1)/((t*x*y^2 + t*x^2 + t - 1)*(x^2 + 1)*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x y^{2} - x^{2} - x - 1\\right)} {\\left(x + 1\\right)}}{{\\left(t x y^{2} + t x^{2} + t x + t - 1\\right)} {\\left(x^{2} + x + 1\\right)} {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x*y^2 - x^2 - x - 1)*(x + 1)/((t*x*y^2 + t*x^2 + t*x + t - 1)*(x^2 + x + 1)*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 5 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x^{2} y^{2} - x^{2} + y^{2} - x - 1\\right)} {\\left(x + 1\\right)}}{{\\left(t x^{2} y^{2} + t x^{2} + t y^{2} + t x + t - 1\\right)} {\\left(x^{2} + x + 1\\right)} {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x^2*y^2 - x^2 + y^2 - x - 1)*(x + 1)/((t*x^2*y^2 + t*x^2 + t*y^2 + t*x + t - 1)*(x^2 + x + 1)*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 5 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x y^{2} - x^{2} - 1\\right)} {\\left(x + 1\\right)}}{{\\left(t x^{2} y + t x y^{2} + t x^{2} + t y + t - 1\\right)} {\\left(x^{2} + 1\\right)} {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x*y^2 - x^2 - 1)*(x + 1)/((t*x^2*y + t*x*y^2 + t*x^2 + t*y + t - 1)*(x^2 + 1)*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 6 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x y^{2} - x^{2} - x - 1\\right)} {\\left(x + 1\\right)}}{{\\left(t x^{2} y + t x y^{2} + t x^{2} + t x + t y + t - 1\\right)} {\\left(x^{2} + x + 1\\right)} {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x*y^2 - x^2 - x - 1)*(x + 1)/((t*x^2*y + t*x*y^2 + t*x^2 + t*x + t*y + t - 1)*(x^2 + x + 1)*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 7 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x^{2} y^{2} - x^{2} + y^{2} - x - 1\\right)} {\\left(x + 1\\right)}}{{\\left(t x^{2} y^{2} + t x^{2} y + t x^{2} + t y^{2} + t x + t y + t - 1\\right)} {\\left(x^{2} + x + 1\\right)} {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x^2*y^2 - x^2 + y^2 - x - 1)*(x + 1)/((t*x^2*y^2 + t*x^2*y + t*x^2 + t*y^2 + t*x + t*y + t - 1)*(x^2 + x + 1)*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Postive drift models\n", "for k in [PD1,PD2,PD3,PD4,PD5,PD6]:\n", " print(\"The generating function for the step set\")\n", " show(plot_steps(k), figsize=2, axes=false)\n", " print(\"is the main power series diagonal of\")\n", " show(RatDiag(k))" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 3 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x^{2} y^{2} + y^{2} - x\\right)} {\\left(x + 1\\right)}}{{\\left(t x^{2} y^{2} + t y^{2} + t x - 1\\right)} x {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x^2*y^2 + y^2 - x)*(x + 1)/((t*x^2*y^2 + t*y^2 + t*x - 1)*x*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x^{2} y^{2} + x y^{2} + y^{2} - x\\right)} {\\left(x + 1\\right)}}{{\\left(t x^{2} y^{2} + t x y^{2} + t y^{2} + t x - 1\\right)} x {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x^2*y^2 + x*y^2 + y^2 - x)*(x + 1)/((t*x^2*y^2 + t*x*y^2 + t*y^2 + t*x - 1)*x*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 5 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x^{2} y^{2} + x y^{2} - x^{2} + y^{2} - 1\\right)} {\\left(x + 1\\right)}}{{\\left(t x^{2} y^{2} + t x y^{2} + t x^{2} + t y^{2} + t - 1\\right)} {\\left(x^{2} + 1\\right)} {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x^2*y^2 + x*y^2 - x^2 + y^2 - 1)*(x + 1)/((t*x^2*y^2 + t*x*y^2 + t*x^2 + t*y^2 + t - 1)*(x^2 + 1)*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 5 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x^{2} y^{2} + y^{2} - x\\right)} {\\left(x + 1\\right)}}{{\\left(t x^{2} y^{2} + t x^{2} y + t y^{2} + t x + t y - 1\\right)} x {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x^2*y^2 + y^2 - x)*(x + 1)/((t*x^2*y^2 + t*x^2*y + t*y^2 + t*x + t*y - 1)*x*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAAB7CAYAAABUx/9/AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjMuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/d3fzzAAAACXBIWXMAAA9hAAAPYQGoP6dpAAAHuUlEQVR4nO2dS0hUXxzHvz7mn6WVEUZN5MbMIlwo0SZ6QOUmatkmWtXCWkTLzFrqBC2sdBFB0KqixzKEXlqt0pUWgaTE9DSih0lqjc7vv7gcZu44c+fcM/ec+zjnAxfmcR8/5jPfc4937jmWEREMelDudwEGdRjZGmFka4SRrRFGtkYY2RphZGuEka0RRrZGVLpY11xqCy5lPCuZZGuEka0R2skeGQF27AAuX/a7EvWUufjVK/TnbCIgHgcmJ63n4+NAQ4O/NXmEOWfnMjCQEQ0A3d3+1eIH2iSbCNi9G3jxIvNaLGalu77ev7o8wiQ7m4EBu2gASKWARMKfevxAi2TnSzUjIuk2yWbkSzVDp3RHPtlOqWZEIN0m2YBzqhm6pDvSyeZJNSMWAyYmgA0b5NclAZNsnlQzdEh3ZJPtJtWMEKdb72S7STUj6umOZLJFUs0Iabr1TbZIqhlRTnckZd+/X9r29+55U0fQiKTsgweB2lqxbcvLgSNHPC0nMETynA0A8/PAv3+LXz90CHjyxHr85QuwYoX9/YoKYMkS+fV5DNc5280Nh6GistJacqmoyDxetsxadCGSzbghP0a2RhjZGmFka4SRrRFGtkYY2RphZGuEka0RRrZGGNkaYWRrhJGtEUa2RhjZGmFka4SRrRFGtkYY2RphZGuEka0RymTPzQG9vUBTE9DaCnz/rurIweP0aWDNGuDECeD9e4UHJiLeRYjZWaIrV4jicSJrFJa13LolusfSaGvL1DA1pf74c3NE5eWZGmIxovZ2omSypN1yOZSWbJbkhgbg1Cng82f7+3//yjpysFlYANLpzPNUCrh6Fdi4UX7SPZddTLIhPyqkeybbSPYGmdJLlm0ky0GGdOGxXrOzQF8f0NNjDZAT2f73b9GjizM/n3nsx/FnZtytz6Rfvw4cPQqcPVvC5Lq8Pbnsrt/kJFFVlb13bRZ1y6VLCnvjg4NW823whzt3xLYTGp+dTgN79wLPnlnfNRG2bgXWrxfbVoR374C3bxe/vmoV0NKSf3ivDBYWMuPDRVi7Fnj4EGhutr3MNT5bqBlnJJPWBYFYzH1TdOOG2+sG4iQSzrXs2kU0Pa2mlj9/xJrunTuJnj4lSqfz7pbLYUmyGSLSVcnOFd3YmHm8cqV64W5lF5HMUCeb4Ua6Ctm5ohMJ++XSgQGi2lq1wnllc0pmqJfN4JEuW3Y+0USLr40PD6sVXky2S8kM/2QznKTLlF1INFH+H0JUCi8kW1Ayw3/ZjHzS+/tL2WNhnEQTFf7VS5XwVMreVyhRMiM4shnJJFFnpyUglfJij3aKiSZy/olTlfD+fqKTJz2RzAiebJnwiCYq/nu26nO4R+gjm1c0Ed/NCyEUrodsN6KJ+O9UCZnw6Mt2K5rI3W1JIRIebdkioonc34MWEuHRlS0qmkjshsMQCI+m7FJEE4nfXRpw4dGTXapootJuJQ6w8GjJ9kI0Uen3jQdUeHRkeyWayJtBAgEUHg3ZXoom8m5ESMCEh1+216KJvB3+EyDh4ZYtQzSR92O9AiI8vLJliSaSM7AvAMLDKVumaCJ5ozh9Fh4+2bJFE8kdsuuj8HDJViGaSP74bJ+Eh0e2KtFEagbj+yA8HLJViiZSN/OCYuHBl61aNJHaaTYUCg+2bD9EE6mfU0WR8ODK9ks0kT8T6CgQzuXQs7GLHz8C09PAli3O6124AHR0ZJ4nEsCZM15VYTE/Dzx+DHz9uvi9T58yj2/eBJYutb//339AWxuwerV39WzbBjx6BOzfD/z6BTx/Dhw4ADx4ANTUFN7uxw9r9GlrK1DGN07TGd5vhdPXanycqLra+tZ2dxdeT1Wiu7vtx3G7tLTIqctNwoeGMuv29RXdtbpm/Nw5+4fV1bV4HZVNd2dnabI3bZJXG4/woSH7qJGmpoCM4kynrWJyP7Bs4arP0cmk2JhxtvT2yq3PSXiuaLaMjjruUo3s0dHCH1pXl3+dsfZ2MdHxuDUro2zyCR8czC8aIDp/3nF3amTnNuFOi8pet2i6Zac6m1zhFRWF6yrSlMuXXagJ91s0w226VaU6m+FhopoavvocmnIuhyVNevf6NTA2xrdu9nydqujoAGIxd+tXVcmrJx9E/OvevVvywcST7aYJz+20qYI33X6kulBnTKApl9uMu2nC/RTOe+5Wea4mci+6SFMuV7ZTL7zY4nThRQbF0q061cPDYqKBgr1yubLdNuG5y5s3nn5+jhRLt+pUNzeLf24FmnIuh0IdNKLSOgvV1cDy5eLbu6W+Hjh2LP978Thw/Li6WgBrlkJRxsasjrEQvN+K7K+RaBO+ebPVDE1MyEmME4XSrTrVRNaMSdeuEe3b5/y3tYumXF4z7qYJZ4JfvfJsshhhcs/d69ap74Hn8u2be/GNjYs+Szmy02miurrwCM4mmSQqK8vUefGi3xXZcSP+5UvbpnJkj4yES3Aue/ZY9VZVEc3M+F1NYYqJP3zYtroc2T9+EFVWWlvW1YVDcDYLC0S3b1sT5IcFJn779ozsnh7bKlwOheYb//kT+PDBmvPakzsoDNxMTgJTU9Y/w8uCy4KQbEPg4JJt/henRri54dA02CHHJFsjjGyNMLI1wsjWCCNbI4xsjTCyNcLI1ggjWyP+B9WBEFr6mCIAAAAAAElFTkSuQmCC\n", "text/plain": [ "Graphics object consisting of 6 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x^{2} y^{2} + x y^{2} + y^{2} - x\\right)} {\\left(x + 1\\right)}}{{\\left(t x^{2} y^{2} + t x^{2} y + t x y^{2} + t y^{2} + t x + t y - 1\\right)} x {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x^2*y^2 + x*y^2 + y^2 - x)*(x + 1)/((t*x^2*y^2 + t*x^2*y + t*x*y^2 + t*y^2 + t*x + t*y - 1)*x*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 7 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{{\\left(x^{2} y^{2} + x y^{2} - x^{2} + y^{2} - 1\\right)} {\\left(x + 1\\right)}}{{\\left(t x^{2} y^{2} + t x^{2} y + t x y^{2} + t x^{2} + t y^{2} + t y + t - 1\\right)} {\\left(x^{2} + 1\\right)} {\\left(y - 1\\right)}}\n", "\\end{math}" ], "text/plain": [ "-(x^2*y^2 + x*y^2 - x^2 + y^2 - 1)*(x + 1)/((t*x^2*y^2 + t*x^2*y + t*x*y^2 + t*x^2 + t*y^2 + t*y + t - 1)*(x^2 + 1)*(y - 1))" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Negative drift models\n", "for k in [ND1,ND2,ND3,ND4,ND5,ND6]:\n", " print(\"The generating function for the step set\")\n", " show(plot_steps(k), figsize=2, axes=false)\n", " print(\"is the main power series diagonal of\")\n", " show(RatDiag(k))" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 3 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(x^{2} - y\\right)} {\\left(x y - 1\\right)} {\\left(y^{2} - x\\right)}}{{\\left(t x^{2} y + t y^{2} + t x - 1\\right)} {\\left(x - 1\\right)} x {\\left(y - 1\\right)} y}\n", "\\end{math}" ], "text/plain": [ "(x^2 - y)*(x*y - 1)*(y^2 - x)/((t*x^2*y + t*y^2 + t*x - 1)*(x - 1)*x*(y - 1)*y)" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 6 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(x^{2} - y\\right)} {\\left(x y - 1\\right)} {\\left(y^{2} - x\\right)}}{{\\left(t x^{2} y + t x y^{2} + t x^{2} + t y^{2} + t x + t y - 1\\right)} {\\left(x - 1\\right)} x {\\left(y - 1\\right)} y}\n", "\\end{math}" ], "text/plain": [ "(x^2 - y)*(x*y - 1)*(y^2 - x)/((t*x^2*y + t*x*y^2 + t*x^2 + t*y^2 + t*x + t*y - 1)*(x - 1)*x*(y - 1)*y)" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The generating function for the step set\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "is the main power series diagonal of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{{\\left(x^{2} - y\\right)} {\\left(x + y\\right)} {\\left(x - y\\right)} {\\left(x + 1\\right)}}{{\\left(t x^{2} y + t x^{2} + t y^{2} + t y - 1\\right)} x^{2} y}\n", "\\end{math}" ], "text/plain": [ "(x^2 - y)*(x + y)*(x - y)*(x + 1)/((t*x^2*y + t*x^2 + t*y^2 + t*y - 1)*x^2*y)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "for k in [SP1,SP2,SP3]:\n", " print(\"The generating function for the step set\")\n", " show(plot_steps(k), figsize=2, axes=false)\n", " print(\"is the main power series diagonal of\")\n", " show(RatDiag(k))" ] }, { "cell_type": "code", "execution_count": 0, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.2", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }