{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# MPI 4 - Cours 2\n", "\n", "## Un peu de calcul non commutatif : séries rationnelles et langages rationnels\n", "\n", "Considérons l'alphabet $A=\\{a,b\\}$, et intéressons nous au langage $L=(a+ba)^*$.\n", "\n", "L'ensemble $C=\\{a,ba\\}$ est ce qu'on appelle un *code préfixe*, c'est à dire que s'il contient un mot\n", "$w$, il ne contient aucun préfixe $u$ de $w$ (un mot tel que $w=uv$ avec $u,v\\not=\\epsilon$).\n", "C'est un *code*, car tout mot de $C^*$ n'a qu'une seule écriture comme concaténation d'éléments de $C$ (unicité du déchiffrage, ou encore, non-ambiguïté de la grammaire) :\n", "$$w=baaabaaababaaa \\rightarrow ba.a.a.ba.a.a.ba.ba.a.a$$\n", "Le langage $L$ est reconnu par l'automate\n", "![automate](automate.png)\n", "Soit $u_n$ le nombre de mots de longueur $n$ dans $L$. On a \n", "$$L = \\{\\epsilon, a, ba, aa, baa, aba, aaa, baba,baaa,abaa,aaba,aaaa,\\cdots\\}$$\n", "et donc\n", "$$u_0=1,\\ u_1=1,\\ u_2=2,\\ u_3=3,\\ u_4=5,\\cdots$$\n", "Si on note $L_n$ l'ensemble des mots de longueur $n$ dans $L$, on voit que pour $n\\ge 2$,\n", "$$L_n=aL_{n-1}+baL_{n-2}$$\n", "puisqu'un mot de $L$ commence soit par $a$, soit par $ba$. Donc, $u_n=u_{n-1}+u_{n-2}$ avec $u_0=1$, $u_1=1$,\n", "c'est donc la suite de Fibonacci décalée de 1.\n", "\n", "Une technique classique pour étudier ce genre de problème est d'identifier les mots à des monômes en variables non commutatives, la concaténation devenant alors la multiplication et son élément neutre $\\epsilon$ le scalaire 1.\n", "Le code $C$ s'identifie alores au polynôme\n", "$$C= a+ba$$\n", "et le langage $L$ s'identifie à la série formelle\n", "$$L = C^* =\\sum_{n\\ge 0}C^n = 1+C+C^2+\\cdots = \\frac1{1-C}=\\frac1{1-a-ba}=1+a+ba+baa+aba+aaa+\\cdots.$$\n", "On trouve alors automatiquement la série génératrice de $u_n$ en remplaçant $a$ et $b$ par $x$ dans cette\n", "expression, car les mots de longueur $n$ deviennent tous $x^n$\n", "$$U(x)=\\frac1{1-x-x^2}=1+x+2x^2+3x^3+5x^4+8x^3+\\cdots$$\n", "C'est donc la suite de Fibonacci décalée $u_n=F_{n+1}.$\n", "\n", "D'une manière générale, l'expression $C^*$ se traduit par $(1-C)^{-1}$, mais on ne peut en déduire le dénombrement des mots par longueur que si l'expression est non ambiguë. Par exemple, $C=a+ab+ba$ n'est pas un code, puisque\n", "$w=aba$ possède deux dérivations : $w=a.ba = ab.a$.\n", "Dans la série non commutative\n", "$$(1-a-ab-ba)^{-1}=1+a+ab+ba+ 2aba+aab+baa+aaa+\\cdots$$\n", "le coefficient d'un mot est le nombre de ses factorisations en éléments de $C$. On peut montrer en construisant l'automate minimal de $C^*$ qu'une expression non ambigue serait\n", "$$((b+a(a+ba)^*bb)a)^*(\\epsilon+a((a+ba)^*(\\epsilon+b))$$\n", "En remplaçant $a$ et $b$ par $x$ et les $X^*$ par des $(1-X)^{-1}$, on trouve pour la série génératrice\n", "$$\n", "\\frac1{1-x-2\\,{x}^{2}+{x}^{3}}=1+x+3\\,{x}^{2}+4\\,{x}^{3}+9\\,{x}^{4}+14\\,{x}^{5}+28\\,{x}^{6}+47\\,{x}^{7}+89\\,{x}^{8}+155\\,{x}^{9}+286\\,{x}^{10}+507\\,{x}^{11}+O \\left( {x}^{12} \\right)$$\n", "Si on avait fait cela avec l'expression ambiguë, on aurait trouvé le résultat faux\n", "$$\\frac1{1-x-2x^2}=1+x+3\\,{x}^{2}+5\\,{x}^{3}+11\\,{x}^{4}+21\\,{x}^{5}+43\\,{x}^{6}+85\\,{x}^{7}+171\\,{x}^{8}+341\\,{x}^{9}+683\\,{x}^{10}+1365\\,{x}^{11}+O \\left( {x}^{12} \\right)$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## La décomposition en éléments simples" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "Lorsque la série génératrice d'une suite représente une fraction rationnelle, on peut la décomposer en éléments simples, pourvu que l'on connaisse les racines du dénominateur : si\n", "$$U(x) = \\frac{A(x)}{B(x)}$$\n", "avec \n", "$$B(x)=(x-\\beta_1)^{m_1}(x-\\beta_2)^{m_2}\\cdots(x-\\beta_n)^{m_n}$$\n", "alors, on peut trouver une décomposition de la forme\n", "$$U(x)=P(x) +\\sum_{k=1}^n\\sum_{i_k=1}^{m_k}\\frac{a_{ik}}{(x-\\beta_k)^{i_k}}$$\n", "où $P(x)$ est un polynôme (qui n'apparaît que si le degré de $A$ est supérieur ou égal à celui de $B$, auquel cas c'est le quotient de $A$ par $B$), et pour chaque racine $\\beta$, on a un élément simple $\\frac1{(x-\\beta)^k}$ pour $k$ de 1 jusqu'à la multiplicité $m$ de $\\beta$." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA8AAAAPCAYAAAA71pVKAAAABHNCSVQICAgIfAhkiAAAAPhJREFUKJGl0i9LREEUh+FndaMfwGYxaNEvoNEVLIJFDGLQYBMEQWWFGxQWNAhGi2KTjRbFbLIYBIM2QcFkMRh2DTPgZXYuwnrKzDnze+f8makVRaFfG0j8G3SxkMRrOItnrSp4Cx3sY7AUP8IKTrFdBT/gAuNYjrFdbOIS62VxPdNKE4soMIQDXMfLOmVhmhlecYwRnOBOmMF3KszB8FHar+IrJ8rBS8KA3qO/UZGgB57DOR4xgSesYewveAptoedGLH1PGGqrF/2FJ3GFT8zgLcbbuMc8pnPwqPAUXcziJdHsxPUwhet4xnCurGi3wvesLLsv+xf8A0vhLIYSfI/nAAAAAElFTkSuQmCC\n", "text/latex": [ "$\\displaystyle x$" ], "text/plain": [ "x" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import *\n", "init_printing()\n", "var('x')" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\frac{2 x^{12} - x^{3} + x + 2}{\\left(1 - 2 x\\right)^{3} \\left(1 - x\\right)^{4} \\left(3 x + 1\\right)^{2}}$" ], "text/plain": [ " 12 3 \n", " 2⋅x - x + x + 2 \n", "──────────────────────────────\n", " 3 4 2\n", "(1 - 2⋅x) ⋅(1 - x) ⋅(3⋅x + 1) " ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Exemple au hasard\n", "U = (2*x**12-x**3+x+2)/((1-2*x)**3*(1+3*x)**2*(1-x)**4);U" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle - \\frac{x^{3}}{36} - \\frac{29 x^{2}}{216} - \\frac{85 x}{216} - \\frac{3485}{3888} + \\frac{1054999}{3888000 \\left(3 x + 1\\right)} + \\frac{45271}{388800 \\left(3 x + 1\\right)^{2}} - \\frac{153003}{4000 \\left(2 x - 1\\right)} - \\frac{1389}{80 \\left(2 x - 1\\right)^{2}} - \\frac{973}{160 \\left(2 x - 1\\right)^{3}} + \\frac{2213}{128 \\left(x - 1\\right)} - \\frac{411}{64 \\left(x - 1\\right)^{2}} + \\frac{1}{2 \\left(x - 1\\right)^{3}} - \\frac{1}{4 \\left(x - 1\\right)^{4}}$" ], "text/plain": [ " 3 2 \n", " x 29⋅x 85⋅x 3485 1054999 45271 15300\n", "- ── - ───── - ──── - ──── + ───────────────── + ───────────────── - ─────────\n", " 36 216 216 3888 3888000⋅(3⋅x + 1) 2 4000⋅(2⋅x\n", " 388800⋅(3⋅x + 1) \n", "\n", " \n", "3 1389 973 2213 411 1 \n", "───── - ───────────── - ────────────── + ─────────── - ─────────── + ─────────\n", " - 1) 2 3 128⋅(x - 1) 2 \n", " 80⋅(2⋅x - 1) 160⋅(2⋅x - 1) 64⋅(x - 1) 2⋅(x - 1)\n", "\n", " \n", " 1 \n", "─ - ──────────\n", "3 4\n", " 4⋅(x - 1) " ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "U.apart()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notons que cette décomposition peut faire intervenir des racines complexes ou irrationnelles. Pour que `sympy` accepte de les calculer (s'il ne doit résoudre que des équations de degré au plus 4), il faut utiliser l'option `full=True` et la méthode `doit`." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{1}{x^{2} + x + 1}$" ], "text/plain": [ " 1 \n", "──────────\n", " 2 \n", "x + x + 1" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "V = 1/(1+x+x**2);V" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{1}{x^{2} + x + 1}$" ], "text/plain": [ " 1 \n", "──────────\n", " 2 \n", "x + x + 1" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "V.apart()" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAARYAAAArCAYAAAC93qeIAAAABHNCSVQICAgIfAhkiAAABtZJREFUeJztnVuoFlUUx38elQQtzSiFBIXCIrIj5EnNvGSU2YVjmN0s6aEXu0APPuTl4RBFF4kulPngg4lUhEQnsxsZoZlJgd3JSulmUdqptESzPD2sPThnvr2/uXwzZ27rB8P5ZmbPzFqz/6zZe589awZ0dXWhKIqSJm15G9CP9LawKEozJqPa6kOdAstdwICEi6I0owPVVh/qElgGAkfzNkKpJKotC3UJLNOA9/I2Qqkkqi0LdQksE4BPWzj+KWBDSrYo1UK1ZaEugSU4SHYH8AlwwCzbgSubHL8CuDUTy5Syo9qyUIfAMh7YFdj2I3APcD4wCXgbeAk4z3GO34G/sjJQKS2qLQdVDCyL6PuEmAlsCZTpBl4Fvga+ApYDB4GplvONQZ5KZ6VuqVI2VFsRqVpgOQH4FbjJt20wzUftBwI3AsOwD8JNBA4hQlHqi2orBlULLEeA14GxwCDgFGC/o+wEpAl6BFgFzMM+CNeO9JmPpW2sUipUWzEYlLcBGbEVmAGchojBxi7kiTECmA+sA2YBnwXKTQQ+ysRKpYyotiJQtRaLRzfQCZwM/OEo8w/wDfAhsBSp4Lst5dqpaOUriVBtRaCqgWUHMAVpikalDRgS2DYUOIOKVr6SCNVWBKraFepF+q7vOPY/CGwCfgBORAbkZtE438D7F2ErE6CUaqHaikBVAwvIXILfHPtGA+vN3z8RocwF3giUa0f+ZXgoIxuVcqLaCmGA5mNRFCVtqjrGorTGt8TLKbI+FyuVwlLlrpCSnN3A4Rjlf8rKEKWcaGBRbFyStwFKufECS2VT5ClN6Y8MZqqtGuIFlsqmyFNyR7VVQ5IO3s5M1QolK8paT2W1u2446ylJYOkAhie3pXbkmcH9GHBZCufpT1Rf0cn76wBOfSUJLIuBl1syp17kmcF9K3At5RqkV31FJ++vAzj1FTewzKWCuSMypAgZ3LcDC3O2ISqqr+gUQVvg0FfcwLKAxoxZipsiZHDfitRbGVB9RacI2gKHvuIGljnAx6mYUw+KkMF9N9JkPqHF8/QHqq/oFEFb4NCXP7CEZRcfirytmVfi3xnARmSWZy/St3OxBnjU/F4KfID4tM+c49zszOxDFhnc4/rWi+RcPT2e6akSxe889aXaEpL4ZtWXP7CEZRcfZS6SNmuBrgjlhiJPsztDyrUBVyEJeUBeWV8FXAjMBv4F3gJGpmibjSwyuCf17QBSf3kRxe8s9LWW4mqrFYqkLbDoyz+a2923LMuREfqpSCQcbE4eZAHyEtp44Duz7XHgCmPUPodjcXnNLGFMRmx916zPCey/BXmdfRoShdNiEfIq/SazPhMRtp+we+xnDJLT42yOiyipb0fNcXkRxe889aXaas23Bn25xlhs2cX3ASdZym5A+norzPoS4AbgctILKnHoRCrAJlKQ5nYb0JPiNfsrg3tS34Yb+4qAy+8y6Eu1ZfetQV/B/z9PQPpmQ5B+kz+7eI8xeFDg4r3AMmPUbvN7tvmdB51ItHbxGJIO8P0Ur+llcF+B3J/hNM/g7rrHfmwZ3JP6NoL830AO87sM+lJt2X1r0FewxeJlF58CrEayi/sHbLYA51hO/CYy0HMfcB2SRNjFMqRv5y0LLdumNzm+GeOBcTRm6/JYiQzULQD+y8A2L4P7pcg9sRF2jz2CGdyT+jYK2EM242NxiOJ3q/oqsra6CJ8NO6vJ9YuoLXDoK9hi8bKLg1ReB5Jd/DazbSNwAY19ttlIFGwjvMm9GnjBt/4QsBd4wrdtb8g5XHQCm4G/LfseAW4GLua4j2nb1o00Qb8kPIM72O+xR7ux2SOpbx0UYyZrFL9b1VeRtfUk8HzINb5vsq+I2gKHvsKmegezi69DKm9NwMgXgduBa4D7kRmULnro2087aNZdFRKHThoHtUCEdT1yc77I0LYdyMBinLkYUTO4J/VtLs2buHlh87tVfRVZW/txd2GiUERtgUNf/sASJbv4YaQZ1o44OBb5Tu1K4BlgpzF4OtJ0S5NhwJm+9XFIk64HifSnIiPb8wPHrUIi7jxTdrTZ7jWN0ySrDO5JfRsLfI77CddfRPU7L32ptpL55tSXf4zFyy6+C2kWdSDRKPhvuGeRQaKRyKDSK8hTBMTxDcADwQulwCREWDvN+krz+16zfjXSD/8lcNxi5EZvBn72LUsysBFkLsEex76o9ziYwT2pbxcBTyf0I02i+g356Eu1lcw3p76SZulvo3jfm+0GtgEP521IBiT1rYj1FIWi2a3asuOsp6SJnopU6R7bgOfyNiIjkvpWxHqKQtHsVm3ZcdaTfldIUZTU0e8KKYqSOhpYFEVJnf8BVYAo8DD35WQAAAAASUVORK5CYII=\n", "text/latex": [ "$\\displaystyle \\frac{\\sqrt{3} i}{3 \\left(x + \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right)} - \\frac{\\sqrt{3} i}{3 \\left(x + \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right)}$" ], "text/plain": [ " √3⋅ⅈ √3⋅ⅈ \n", "──────────────── - ────────────────\n", " ⎛ 1 √3⋅ⅈ⎞ ⎛ 1 √3⋅ⅈ⎞\n", "3⋅⎜x + ─ + ────⎟ 3⋅⎜x + ─ - ────⎟\n", " ⎝ 2 2 ⎠ ⎝ 2 2 ⎠" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "V.apart(full=True).doit()" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle \\frac{- \\frac{9}{31 \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}} + \\frac{6 \\left(- \\frac{\\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}{3} + \\frac{1}{\\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}\\right)^{2}}{31} + \\frac{4}{31} + \\frac{3 \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}{31}}{x - \\frac{1}{\\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}} + \\frac{\\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}{3}} + \\frac{\\frac{4}{31} + \\frac{6 \\left(- \\frac{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}{3} + \\frac{1}{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}\\right)^{2}}{31} - \\frac{9}{31 \\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}} + \\frac{3 \\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}{31}}{x - \\frac{1}{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}} + \\frac{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}{3}} + \\frac{\\frac{4}{31} + \\frac{3 \\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}{31} - \\frac{9}{31 \\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}} + \\frac{6 \\left(\\frac{1}{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}} - \\frac{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}{3}\\right)^{2}}{31}}{x + \\frac{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}{3} - \\frac{1}{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27}{2} + \\frac{3 \\sqrt{93}}{2}}}}$" ], "text/plain": [ " 2 \n", " ⎛ ____________ ⎞ \n", " ⎜ ╱ 27 3⋅√93 ⎟ \n", " ⎜ 3 ╱ ── + ───── ⎟ \n", " ⎜ ╲╱ 2 2 1 ⎟ \n", " 6⋅⎜- ──────────────── + ────────────────⎟ \n", " ⎜ 3 ____________⎟ \n", " ⎜ ╱ 27 3⋅√93 ⎟ \n", " ⎜ 3 ╱ ── + ───── ⎟ 3⋅3 \n", " 9 ⎝ ╲╱ 2 2 ⎠ 4 ╲╱\n", "- ─────────────────── + ────────────────────────────────────────── + ── + ────\n", " ____________ 31 31 \n", " ╱ 27 3⋅√93 \n", " 31⋅3 ╱ ── + ───── \n", " ╲╱ 2 2 \n", "──────────────────────────────────────────────────────────────────────────────\n", " ____________ \n", " ╱ 27 3⋅√93 \n", " 3 ╱ ── + ───── \n", " 1 ╲╱ 2 2 \n", " x - ──────────────── + ──────────────── \n", " ____________ 3 \n", " ╱ 27 3⋅√93 \n", " 3 ╱ ── + ───── \n", " ╲╱ 2 2 \n", "\n", " \n", " ⎛ ____________ \n", " ⎜ ⎛ 1 √3⋅ⅈ⎞ ╱ 27 3⋅√93 \n", " ⎜ ⎜- ─ + ────⎟⋅3 ╱ ── + ───── \n", " ⎜ ⎝ 2 2 ⎠ ╲╱ 2 2 1 \n", " 6⋅⎜- ───────────────────────────── + ───────────────────\n", " ____________ ⎜ 3 __\n", " ╱ 27 3⋅√93 ⎜ ⎛ 1 √3⋅ⅈ⎞ ╱ 2\n", "╱ ── + ───── ⎜ ⎜- ─ + ────⎟⋅3 ╱ ─\n", " 2 2 4 ⎝ ⎝ 2 2 ⎠ ╲╱ 2\n", "────────────── ── + ────────────────────────────────────────────────────────\n", " 31 31 31 \n", " \n", " \n", " \n", "────────────── + ─────────────────────────────────────────────────────────────\n", " \n", " \n", " \n", " 1 \n", " x - ───────────────────\n", " __\n", " ⎛ 1 √3⋅ⅈ⎞ ╱ 2\n", " ⎜- ─ + ────⎟⋅3 ╱ ─\n", " ⎝ 2 2 ⎠ ╲╱ 2\n", "\n", " 2 \n", " ⎞ \n", " ⎟ \n", " ⎟ \n", " ⎟ \n", "──────────⎟ \n", "__________⎟ _________\n", "7 3⋅√93 ⎟ ⎛ 1 √3⋅ⅈ⎞ ╱ 27 3⋅√\n", "─ + ───── ⎟ 3⋅⎜- ─ + ────⎟⋅3 ╱ ── + ───\n", " 2 ⎠ 9 ⎝ 2 2 ⎠ ╲╱ 2 2\n", "──────────── - ──────────────────────────────── + ────────────────────────────\n", " ____________ 31 \n", " ⎛ 1 √3⋅ⅈ⎞ ╱ 27 3⋅√93 \n", " 31⋅⎜- ─ + ────⎟⋅3 ╱ ── + ───── \n", " ⎝ 2 2 ⎠ ╲╱ 2 2 \n", "──────────────────────────────────────────────────────────────────────────────\n", " ____________ \n", " ⎛ 1 √3⋅ⅈ⎞ ╱ 27 3⋅√93 \n", " ⎜- ─ + ────⎟⋅3 ╱ ── + ───── \n", " ⎝ 2 2 ⎠ ╲╱ 2 2 \n", "────────── + ───────────────────────────── \n", "__________ 3 \n", "7 3⋅√93 \n", "─ + ───── \n", " 2 \n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", "___ ____________ \n", "93 ⎛ 1 √3⋅ⅈ⎞ ╱ 27 3⋅√93 \n", "── 3⋅⎜- ─ - ────⎟⋅3 ╱ ── + ───── \n", " 4 ⎝ 2 2 ⎠ ╲╱ 2 2 9 \n", "─── ── + ─────────────────────────────── - ──────────────────────────────── \n", " 31 31 ____________ \n", " ⎛ 1 √3⋅ⅈ⎞ ╱ 27 3⋅√93 \n", " 31⋅⎜- ─ - ────⎟⋅3 ╱ ── + ───── \n", " ⎝ 2 2 ⎠ ╲╱ 2 2 \n", "─── + ────────────────────────────────────────────────────────────────────────\n", " ____________ \n", " ⎛ 1 √3⋅ⅈ⎞ ╱ 27 3⋅√93 \n", " ⎜- ─ - ────⎟⋅3 ╱ ── + ───── \n", " ⎝ 2 2 ⎠ ╲╱ 2 2 \n", " x + ───────────────────────────── -\n", " 3 \n", " \n", " \n", " \n", "\n", " 2\n", " ⎛ ____________⎞ \n", " ⎜ ⎛ 1 √3⋅ⅈ⎞ ╱ 27 3⋅√93 ⎟ \n", " ⎜ ⎜- ─ - ────⎟⋅3 ╱ ── + ───── ⎟ \n", " ⎜ 1 ⎝ 2 2 ⎠ ╲╱ 2 2 ⎟ \n", " 6⋅⎜───────────────────────────── - ─────────────────────────────⎟ \n", " ⎜ ____________ 3 ⎟ \n", " ⎜⎛ 1 √3⋅ⅈ⎞ ╱ 27 3⋅√93 ⎟ \n", " ⎜⎜- ─ - ────⎟⋅3 ╱ ── + ───── ⎟ \n", " ⎝⎝ 2 2 ⎠ ╲╱ 2 2 ⎠ \n", "+ ──────────────────────────────────────────────────────────────────\n", " 31 \n", " \n", " \n", " \n", "────────────────────────────────────────────────────────────────────\n", " \n", " \n", " \n", " 1 \n", " ───────────────────────────── \n", " ____________ \n", " ⎛ 1 √3⋅ⅈ⎞ ╱ 27 3⋅√93 \n", " ⎜- ─ - ────⎟⋅3 ╱ ── + ───── \n", " ⎝ 2 2 ⎠ ╲╱ 2 2 " ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Avec des équations du 3ème degré, le résultat n'est pas très exploitable ...\n", "www=1/(1+x+x**3)\n", "www.apart(full=True).doit()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## La formule du binôme généralisé\n", "\n", "On voit qu'il faut être capable de développer en série entière les fractions simples \n", "$$\\frac1{(1-x)^k}.$$\n", "Pour $k=1$, c'est fait. On peut traiter $k=2$ en l'élevant au carré. En utilisant la formule générale de produit\n", "$$\\sum_{p\\ge 0}a_px^p\\sum_{q\\ge 0}b_qx^q = \\sum_{n\\ge 0}\\left(\\sum_{p=0}^na_pb_{n-p}\\right)x^n$$\n", "on a\n", "$$\\left(\\frac1{1-x}\\right)^2 = \\sum_{n\\ge 0}\\left(\\sum_{p=0}^n 1\\times 1\\right)x^n=\\sum_{n\\ge 0}(n+1)x^n$$\n", "Pour peu que l'on sache que \n", "$$1+2+\\cdots +n = \\frac{n(n+1)}{2}$$\n", "on peut remultiplier par $\\frac1{1-x}$ et obtenir\n", "$$\\left(\\frac1{1-x}\\right)^3 = \\sum_{n\\ge 0}\\left(\\sum_{p=0}^n (p+1)\\times 1\\right)x^n=\\sum_{n\\ge 0}\\frac{(n+1)(n+2)}2x^n$$\n", "Avec un effort supplémentaire, on peut encore trouver $k=4$ et deviner la formule générale\n", "$$\\left(\\frac1{1-x}\\right)^k = \\sum_{n\\ge 0}{n+k-1\\choose n}x^n$$\n", "et la prouver par récurrence.\n", "\n", "Ou encore, la prouver combinatoirement : si on fait le produit de $k$ séries géométriques\n", "$$\\frac1{(1-x_1)(1-x_2)\\cdots (1-x_k)}=\\sum_{n_1\\ge 0}x_1^{n_1}\\sum_{n_2\\ge 0}x_2^{n_2}\\cdots\\sum_{n_k\\ge 0}x_k^{n_k},$$\n", "on voit que le membre droit est la somme de tous les monômes en $x_1,\\ldots,x_k$. Si l'on pose\n", "$$x_1=x_2=\\cdots=x_k=x$$\n", "dans cette égalité, le membre gauche devient $\\left(\\frac1{1-x}\\right)^k$, et le coefficient de $x^n$ dans le membre droit est le nombre de monômes de degré total $n$ que l'on peut former avec $k$ variables, c'est à dire, le nombre de sélections avec répétitions de $n$ objets parmi $k$, connu pour être égal à ${n+k-1\\choose n}x^n$ d'après le cours de L2.\n" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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rKkmV71AUtLsWTUZ2ps2jaP+z09A2IC9SPLqiwOgXUIf6GgVZHocyQFeSnkVXijIUiqTL2hsF685BZvR1KDi8O7IazETmfZukyzsIbbFyDYpdfA0puT1RH+1Hyn2SdFmd9LeOfr/UkXRZH0X79HVGrqxpaFxqiSa+MtTO7Z/oSrq8oC1qNqLQoXVoEdcNeS56kbLUVwdZw8zrA5CSMw4ZXBajvt8JuXivcVyb5Gc0Grn2z0GWuMeQJXAvtOBphfpCT/wTuqaiX3vpQvoPJ5TKczkJjc/bXNRuxa81UracNCO1t9EnFFfxs/dZ8tot2+Z51ID3RwHmW1EFLHVdNxh19lvJ3HiykMxGE8YxqCHURyulJSgZYhypJJVSlaEQVAdZ5yAz+mFoIK2DFJ+XUN0+SGqvveog75do8B+KBsKj0KT5JMpYs10/1UFWm5Zo1e6X1FEdZN2ClJ5LgDNQ3e6CxqWZaIyys1urg7ygyfcMNOHXRj8ddy+ywlRa11QXWcPM60vRtiI3IuNFN6S4jENWv+oyZ21B7v3pKOGiK5qj16BYx0EovOO7LPeYihI2zyW1B2epPJd6SOmfgcNbkW07F4PBYDAYDAZDduytXtqQ/ju+cXMpUtY74LAsJiHGz2AwGAwGg6FUGY1cxKW0QXdtpJBOxeVONoqfwWAwGAwGQ3g2IZfxfPy38Co2TVCYWUZ4XlJ/ss1gMBgMBoOhVHjBepUKi/H5yVZj8TMYDAaDwWDYTjCKn8FgMBgMBsN2wv8BaIZzxveLSbQAAAAASUVORK5CYII=\n", "text/latex": [ "$\\displaystyle 1 + 2 x + 3 x^{2} + 4 x^{3} + 5 x^{4} + 6 x^{5} + 7 x^{6} + 8 x^{7} + 9 x^{8} + 10 x^{9} + O\\left(x^{10}\\right)$" ], "text/plain": [ " 2 3 4 5 6 7 8 9 ⎛ 10⎞\n", "1 + 2⋅x + 3⋅x + 4⋅x + 5⋅x + 6⋅x + 7⋅x + 8⋅x + 9⋅x + 10⋅x + O⎝x ⎠" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "series(1/(1-x)**2,x,0,10)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle 1 + 3 x + 6 x^{2} + 10 x^{3} + 15 x^{4} + 21 x^{5} + 28 x^{6} + 36 x^{7} + 45 x^{8} + 55 x^{9} + O\\left(x^{10}\\right)$" ], "text/plain": [ " 2 3 4 5 6 7 8 9 ⎛ 10\n", "1 + 3⋅x + 6⋅x + 10⋅x + 15⋅x + 21⋅x + 28⋅x + 36⋅x + 45⋅x + 55⋅x + O⎝x \n", "\n", "⎞\n", "⎠" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "series(1/(1-x)**3,x,0,10)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle 1 + 4 x + 10 x^{2} + 20 x^{3} + 35 x^{4} + 56 x^{5} + 84 x^{6} + 120 x^{7} + 165 x^{8} + 220 x^{9} + O\\left(x^{10}\\right)$" ], "text/plain": [ " 2 3 4 5 6 7 8 9 \n", "1 + 4⋅x + 10⋅x + 20⋅x + 35⋅x + 56⋅x + 84⋅x + 120⋅x + 165⋅x + 220⋅x + O\n", "\n", "⎛ 10⎞\n", "⎝x ⎠" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "series(1/(1-x)**4,x,0,10)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On peut faire encore mieux. Rappelons qu'on définit $x^{p/q}$ comme $\\sqrt[q]{x^p}$, et qu'on peut définir $x^\\alpha$ pour $\\alpha$ réel quelconque comme $\\exp(\\alpha\\ln x)$.\n", "\n", "On peut donc donner un sens à la série formelle $(1+x)^\\alpha$ pour $\\alpha$ quelconque. Posons\n", "$$(1+x)^\\alpha = \\sum_{n\\ge 0}a_nx^n$$\n", "et cherchons les coefficients $a_n$.\n", "\n", "On *définit* la dérivation des séries formelles en posant $(x^n)'=\\frac{d}{dx}x^n=nx^{n-1}$ et en prolongeant par linéarité.\n", "Il n'est pas difficile de monter que cette opération vérifie les propriétés usuelles de la dérivée des fonctions.\n", "Par exemple, pour montrer $(fg)'=f'g+fg'$, on commence par le vérifier pour $f=x^p$ et $g=x^q$. Puis on utilise un première fois la linéarité pour en déduire que c'est encore vrai pour $f=x^p$ et $g$ quelconque, puis une seconde fois\n", "pour $f$ et $g$ quelconques.\n", "\n", "On en déduit ensuite par récurrence $(f^n)'=nf^{n-1}f'$, puis par linéarité, $g(f(x))'=g'(f(x))f'(x)$.\n", "\n", "Comme $(1+x)^\\alpha = \\exp(\\alpha\\ln(1+x))$, on a\n", "$$\\frac{d}{dx}(1+x)^\\alpha = \\exp(\\alpha\\ln(1+x))\\alpha\\frac1{1+x}=\\alpha(1+x)^{\\alpha-1},$$\n", "comme on pouvait s'y attendre.\n", "\n", "On peut maintenant écrire\n", "$$(1+x)\\frac{d}{dx}(1+x)^\\alpha = (1+x)\\sum_{n\\ge 0}n a_nx^{n-1}=\\alpha (1+x)^\\alpha=\\sum_{n\\ge 0}\\alpha a_nx^n.$$\n", "En effectuant le produit par $1+x$, on trouve\n", "$$\\sum_{n\\ge 0}(n+1)a_{n+1}x^n+\\sum_{n\\ge 0}na_nx^n=\\sum_{n\\ge 0}\\alpha a_nx^n$$\n", "donc, en identifiant les coefficients de $x^n$\n", "$$(n+1)a_{n+1}=(\\alpha-n)a_n,\\ \\text{d'où}\\ a_{n+1}=\\frac{\\alpha-n}{n+1} \n", "\\frac{\\alpha-n+1}{n} \\frac{\\alpha-n+2}{n-1}\\cdots \\frac{\\alpha}{1}a_0. $$\n", "Comme on a clairement $a_0=1$, on a finalement\n", "$$a_n = \\frac{\\alpha(\\alpha-1)\\cdots(\\alpha-n+1)}{n!} =: {\\alpha\\choose n}.$$\n", "Ces nombres sont appeliés coefficients binomiaux généralisés. Quand $\\alpha$ est un entier positif, ils coincident\n", "évidemment avec les coefficients binomiaux ordinaires.\n", "\n", "\n", "### Exemple : $\\alpha=\\frac12$\n", "\n", "On a\n", "$\\def\\a{\\frac12}$\n", "$${1/2\\choose n}=\\frac{\\a(\\a-1)(\\a-2)\\cdots(\\a-n+1)}{n!}\n", "=\\frac{1(1-2)(1-4)\\cdots(1-2n+2)}{2^nn!}\n", "=(-1)^{n-1}\\frac{1\\cdot 3\\cdot 5\\cdots (2n-3)}{2^nn!}$$\n", "$$= (-1)^{n-1}\\frac{(2n-2)!}{2^{n-1}(n-1)!2^nn!}= \\frac{(-1)^{n-1}}{2^{2n-1}}\\frac1{n}{2n-2\\choose n-1}.$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Application : dénombrement des arbres binaires\n", "\n", "Un *arbre binaire complet*, c'est soit un sommet unique `o`, soit\n", "
\n",
    "    o\n",
    "   / \\\n",
    "  A   B\n",
    "
\n", "\n", "où `A` et `B` sont des arbres binaires complets. Soit $c_n$ le nombre d'arbre binaires complets à $n$ feuilles.\n", "Les premiers exemples sont\n", "\n", "
\n",
    "o         o                             o            o\n",
    "         / \\                           / \\          / \\\n",
    "        o   o                         o   o        o   o\n",
    "                                     / \\              / \\\n",
    "                                    o   o            o   o\n",
    "                                    \n",
    "\n",
    "    _o__           __o___           __o__           __o___           _o__    \n",
    "   /    \\         /      \\         /     \\         /      \\         /    \\   \n",
    "  o     _o_      o       _o_      o       o      _o_       o      _o_     o  \n",
    "       /   \\            /   \\    / \\     / \\    /   \\            /   \\       \n",
    "      o     o          o     o  o   o   o   o  o     o          o     o      \n",
    "           / \\        / \\                           / \\        / \\           \n",
    "          o   o      o   o                         o   o      o   o          \n",
    " \n",
    "\n",
    "\n",
    "
\n", "\n", "On a donc $c_0=0$, $c_1=1$, $c_2=1$, $c_3=2$, $c_4=5$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Comment calculer les $c_n$ ? \n", "\n", "Un arbre à $n$ feuilles aura un sous-arbre gauche $A$ à $p$ feuilles et un sous-arbre droit $B$ à $q$ feuilles,\n", "avec $p+q=n$. Il y a $c_pc_q$ choix pour ces sous-arbres, et $p,q$ sont au moins égaux à 1 et au plus à $n-1$.\n", "Donc, pour $n\\ge 2$,\n", "$$c_n = \\sum_{p=1}^{n-1}c_pc_{n-p} =c_1c_{n-1}+c_2c_{n-2}+\\cdots+c_{n-1}c_1.$$\n", "C'est une récurrence qui fait intervenir *toutes* les valeurs précédentes. Testons là :\n", "$$c_2 = c_1c_1 = 1,$$\n", "$$c_3=c_1c_2+c_2c_1=1+1=2,$$\n", "$$c_4=c_1c_3+c_2c_2+c_3c_1=2+1+2=5,$$\n", "$$c_5=c_1c_4+c_2c_3+c_3c_2+c_4c_1=5+2+2+5=14,$$\n", "$$c_6=c_1c_5+c_2c_4+c_3c_3+c_4c_2+c_5c_1=14+5+4+5+14=42, \\ldots$$\n", "Ces nombres sont très célèbres. On les appelle [nombres de Catalan](https://fr.wikipedia.org/wiki/Nombre_de_Catalan), bien qu'ils aient été découverts par Euler qui les appelait lui-même *nombres de Segner*.\n", "Ce [livre](https://books.google.fr/books/about/Catalan_Numbers.html?id=i5QSBwAAQBAJ&redir_esc=y) de Richard Stanley en recense 214 interprétations différentes.\n", "\n", "Formons la série génératrice\n", "$$C(x)=\\sum_{n\\ge 0}c_n x^n = 0 +x+x^2+2x^3+5x^4+14x^5+42x^6+\\cdots$$\n", "Puisque $c_0=0$, la récurrence s'écrit aussi bien\n", "$$c_n = \\sum_{p=0}^{n}c_pc_{n-p} \\quad\\text{pour $n\\ge 2$} $$\n", "Le membre droit est le coefficient de $x^n$ dans $C(x)^2$. Comme $C(x)$ commence par $x$, \n", "$C(x)^2$ commence par $x^2$, et on a donc en complétant \n", "$$C(x) = x + C(x)^2.$$\n", "Autrement dit, $C(x)$ est solution de l'équation du second degré\n", "$$X^2-X+x=0,$$\n", "de discriminant $1-4x$, de sorte que\n", "$$C(x)=\\frac{1\\pm\\sqrt{1-4x}}{2}$$\n", "On sait que $\\sqrt{1-4x}=1-\\frac12 x+O(x^2)$ par la formule du binôme généralisé, il faut donc choisir la solution avec le signe moins si l'on veut que son premier terme soit $x$\n", "$$C(x)=\\frac{1-\\sqrt{1-4x}}{2}= \\frac12\\left[1-\\sum_{n\\ge 0}{1/2\\choose n}(-4x)^n\\right]\n", "=\\sum_{n\\ge 1}\\frac{(-1)^{n}}{2^{2n}}\\frac1{n}{2n-2\\choose n-1}(-4x)^n\n", "=\\sum_{n\\ge 1}\\frac1{n}{2n-2\\choose n-1}x^n$$\n", "ce qui nous donne finalement\n", "$$c_n = \\frac1{n}{2n-2\\choose n-1}.$$\n", "Vérifions :\n", "$$c_6 = \\frac16{10\\choose 5}=\\frac16 252 = 42.$$" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\left[ 1, \\ 1, \\ 2, \\ 5, \\ 14, \\ 42, \\ 132, \\ 429, \\ 1430, \\ 4862, \\ 16796\\right]$" ], "text/plain": [ "[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796]" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "[binomial(2*n-2,n-1)//n for n in range(1,12)]" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 2 }