OtjddlmZddlmZddlmZmZmZmZddl m Z ddl m Z m Z mZddlmZddlmZmZmZddlmZdd lmZdd lmZdd lmZdd lmZdd l m!Z"m#Z#m$Z%Gdde Z&Gdde&Z!Gdde&Z'Gdde&Z(Gdde&Z)Gdde&Z*Gdde&Z+e*Z,e+Z-Gdde&Z.dS)) annotations)reduce)SsympifyDummyMod)cacheit)DefinedFunctionArgumentIndexError PoleError) fuzzy_and)IntegerpiI)Eq)gmpy)sieve) binomial_mod)Poly) factorialprodsqrtceZdZdZdZdS)CombinatorialFunctionz(Base class for combinatorial functions. c ~ddlm}||}|d}|||d||zkr|S|S)Nr)combsimpmeasureratio)sympy.simplify.combsimpr)selfkwargsrexprrs f/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/functions/combinatorial/factorials.py_eval_simplifyz$CombinatorialFunction._eval_simplifys[444444x~~# 74==F7OGGDMM9 9 9K N)__name__ __module__ __qualname____doc__r$r%r#rrs)22r%rceZdZUdZddZgdZgZded<edZ edZ ed Z d Z d Z dd ZdZdZdZdZdZdZdZdS)raImplementation of factorial function over nonnegative integers. By convention (consistent with the gamma function and the binomial coefficients), factorial of a negative integer is complex infinity. The factorial is very important in combinatorics where it gives the number of ways in which `n` objects can be permuted. It also arises in calculus, probability, number theory, etc. There is strict relation of factorial with gamma function. In fact `n! = gamma(n+1)` for nonnegative integers. Rewrite of this kind is very useful in case of combinatorial simplification. Computation of the factorial is done using two algorithms. For small arguments a precomputed look up table is used. However for bigger input algorithm Prime-Swing is used. It is the fastest algorithm known and computes `n!` via prime factorization of special class of numbers, called here the 'Swing Numbers'. Examples ======== >>> from sympy import Symbol, factorial, S >>> n = Symbol('n', integer=True) >>> factorial(0) 1 >>> factorial(7) 5040 >>> factorial(-2) zoo >>> factorial(n) factorial(n) >>> factorial(2*n) factorial(2*n) >>> factorial(S(1)/2) factorial(1/2) See Also ======== factorial2, RisingFactorial, FallingFactorial cddlm}m}|dkr4||jddz|d|jddzzSt ||)Nr)gamma polygammar,)'sympy.functions.special.gamma_functionsr.r/argsr )r argindexr.r/s r#fdiffzfactorial.fdiffUsjNNNNNNNN q==51)**99Q ! q8H+I+II I$T844 4r%)!r,r,r,r4#r7i;?ii i#r:iSi{/i!imiisXiUiP ioikiIi/LiSi}#r;z list[int]_small_factorialsc|dkr |j|Stt|g}}tjd|dzD]>}d|}} ||z}|dkr|dzdkr||z}nn|dkr||?tj|dz|dzdzD]#}||zdzdkr||$t tj|dzdz|dz}t |}||zS)N!r4r,Tr) _small_swingint_sqrtr primerangeappendr) clsnNprimesprimepq L_product R_products r#_swingzfactorial._swingds6 r66#A& &E!HH rvA)!QU33 % %!1%KA1uuq5A::JAq55MM!$$$)!a%A:: ) )J!#q((MM%(((U-adQhA>>??IV IY& &r%ct|dkrdS||dzdz||zS)Nr?r,) _recursiverN)rErFs r#rPzfactorial._recursives: q551NN1a4((!+SZZ]]: :r%cNt|}|jr |jr tjS|tjur tjS|jr|jr tjS|j }|dkrL|j s4d}tddD]!}||z}|j |"|j |dz }n\ttj|}n@t|d}||d||z zz}t%|SdSdS)Nr,1r?)r is_Numberis_zerorOneInfinity is_Integer is_negativeComplexInfinityrJr<rangerD_gmpyfacbincountrPr)rErFresultibitss r#evalzfactorial.evals7 AJJ ;# +y" +u ajz! +=+,,A2vv"4E%&F%*1b\\EE &!  # 5 < > !3t<===r%Nr,T)r&r'r(r)r3r@r<__annotations__ classmethodrNrPrcrmr|rrrrrrrrr*r%r#rr$sE..`5555L $&%%%%''['<;;[; &+&+[&+P<"""<))) *** ***  >>>>>r%rceZdZdS)MultiFactorialN)r&r'r(r*r%r#rrsDr%rczeZdZdZeedZedZdZdZ dZ d dZ d Z d Z d Zd S) subfactorialaThe subfactorial counts the derangements of $n$ items and is defined for non-negative integers as: .. math:: !n = \begin{cases} 1 & n = 0 \\ 0 & n = 1 \\ (n-1)(!(n-1) + !(n-2)) & n > 1 \end{cases} It can also be written as ``int(round(n!/exp(1)))`` but the recursive definition with caching is implemented for this function. An interesting analytic expression is the following [2]_ .. math:: !x = \Gamma(x + 1, -1)/e which is valid for non-negative integers `x`. The above formula is not very useful in case of non-integers. `\Gamma(x + 1, -1)` is single-valued only for integral arguments `x`, elsewhere on the positive real axis it has an infinite number of branches none of which are real. References ========== .. [1] https://en.wikipedia.org/wiki/Subfactorial .. [2] https://mathworld.wolfram.com/Subfactorial.html Examples ======== >>> from sympy import subfactorial >>> from sympy.abc import n >>> subfactorial(n + 1) subfactorial(n + 1) >>> subfactorial(5) 44 See Also ======== factorial, uppergamma, sympy.utilities.iterables.generate_derangements c|s tjS|dkr tjSd\}}td|dzD]}||dz ||zz}}|S)Nr,)r,rr?)rrVrur[)r rFz1z2ras r#_evalzsubfactorial._evalGsd 5L !VV6MFB1a!e__ / /a!eb2g.BIr%c|jrW|jr|jr||S|tjur tjS|tjurtjSdSdSN)rTrXrrrrNaNrW)rErs r#rczsubfactorial.evalTsi = "~ "#"4 "yy~~%u  ""z!  " " #"r%cV|jdjr|jdjrdSdSdSr)r1is_oddrrrs r#rzsubfactorial._eval_is_even^s< 9Q<  49Q<#> 4    r%cV|jdjr|jdjrdSdSdSrrrs r#rzsubfactorial._eval_is_integerbrr%c ddlm}td}tj|zt |z }t ||||d|fzS)Nr) summationra)sympy.concrete.summationsrrr NegativeOner)r rr!rrafs r#_eval_rewrite_as_factorialz'subfactorial._eval_rewrite_as_factorialfsY777777 #JJ M1 y|| +~~ !aC[ 9 999r%Tc ddlm}ddlm}m}t j|dzz|t tz|zz||dzdz||dzz|dzS)Nr)exp)r. lowergammar,ro) &sympy.functions.elementary.exponentialrr0r.rrrrr)r rrr!rr.rs r#rz#subfactorial._eval_rewrite_as_gammals>>>>>>OOOOOOOO a(aRU3Y7 37B8O8OO%a..!"%#b''* *r%c Fddlm}||dzdtjz S)Nr) uppergammar,ro)r0rrExp1)r rr!rs r#_eval_rewrite_as_uppergammaz(subfactorial._eval_rewrite_as_uppergammars1FFFFFFz#'2&&qv--r%cV|jdjr|jdjrdSdSdSrrrs r#_eval_is_nonnegativez!subfactorial._eval_is_nonnegativevrr%cV|jdjr|jdjrdSdSdSr)r1is_evenrrrs r# _eval_is_oddzsubfactorial._eval_is_oddzs< 9Q<  DIaL$? 4    r%Nr)r&r'r(r)rr rrcrrrrrrrr*r%r#rrs''R    W[ ""["::: **** ...r%rcHeZdZdZedZdZdZdZdZ d dZ d S) factorial2aAThe double factorial `n!!`, not to be confused with `(n!)!` The double factorial is defined for nonnegative integers and for odd negative integers as: .. math:: n!! = \begin{cases} 1 & n = 0 \\ n(n-2)(n-4) \cdots 1 & n\ \text{positive odd} \\ n(n-2)(n-4) \cdots 2 & n\ \text{positive even} \\ (n+2)!!/(n+2) & n\ \text{negative odd} \end{cases} References ========== .. [1] https://en.wikipedia.org/wiki/Double_factorial Examples ======== >>> from sympy import factorial2, var >>> n = var('n') >>> n n >>> factorial2(n + 1) factorial2(n + 1) >>> factorial2(5) 15 >>> factorial2(-1) 1 >>> factorial2(-5) 1/3 See Also ======== factorial, RisingFactorial, FallingFactorial cR|jr|jstd|jrC|jr|dz }d|zt |zSt |t |dz z S|jr)|tj d|z dz zzt | z StddS)Nz ? ">??? ! <;/aAa4)A,,.. ~~ 37(;(;;;z MAMa#gq[99Jt>> from sympy import rf, Poly >>> from sympy.abc import x >>> rf(x, 0) 1 >>> rf(1, 5) 120 >>> rf(x, 5) == x*(1 + x)*(2 + x)*(3 + x)*(4 + x) True >>> rf(Poly(x**3, x), 2) Poly(x**6 + 3*x**5 + 3*x**4 + x**3, x, domain='ZZ') Rewriting is complicated unless the relationship between the arguments is known, but rising factorial can be rewritten in terms of gamma, factorial, binomial, and falling factorial. >>> from sympy import Symbol, factorial, ff, binomial, gamma >>> n = Symbol('n', integer=True, positive=True) >>> R = rf(n, n + 2) >>> for i in (rf, ff, factorial, binomial, gamma): ... R.rewrite(i) ... RisingFactorial(n, n + 2) FallingFactorial(2*n + 1, n + 2) factorial(2*n + 1)/factorial(n - 1) binomial(2*n + 1, n + 2)*factorial(n + 2) gamma(2*n + 2)/gamma(n) See Also ======== factorial, factorial2, FallingFactorial References ========== .. [1] https://en.wikipedia.org/wiki/Pochhammer_symbol .. [2] Peter Paule, "Greatest Factorial Factorization and Symbolic Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268, 1995. c tt|}tjus|tjur tjStjurt |S|jr|jr tjS|jrtjur tjStj ur|j r tj StjSttrWj }t|dkrtdt!fdt#t%|dSt!fdt#t%|dStjur tjStj ur tjSttrkj }t|dkrtddt!fdt#dt't%|dzdz Sdt!fdt#dt't%|dzdz S|jdkrjrjrtjSdSdSdS)Nr,0rf only defined for polynomials on one generatorc4||zSrshiftrrars r#z&RisingFactorial.eval..Qs./nr%c||zzSrr*rs r#rz&RisingFactorial.eval..Uq!a%yr%c6|| zSrrrs r#rz&RisingFactorial.eval..ds01177A2;;r%c||z zSrr*rs r#rz&RisingFactorial.eval..hs,-q1uIr%F)rrrrVrrXrUrrWNegativeInfinityr isinstancergenslenrrr[rArsrqrYrurErrrs ` r#rczRisingFactorial.eval5s AJJ AJJ ::ae5L !%ZZQ<<  \, Jy+ Ju =(JAJ z)a0008.#$#55#$:-%a.. <#$6D"4yy1}}&02K'L'L!L(./=/=/=/=.3CFFmmQ(@(@!@$**@*@*@*@*/A--$<$<<AJ z)a000 z)%a.. 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It is defined by .. math:: \texttt{ff(x, k)} = (x)_k = x \cdot (x-1) \cdots (x-k+1) where `x` can be arbitrary expression and `k` is an integer. For more information check "Concrete mathematics" by Graham, pp. 66 or [1]_. When `x` is a `~.Poly` instance of degree $\ge 1$ with single variable, `(x)_k = x(y) \cdot x(y-1) \cdots x(y-k+1)`, where `y` is the variable of `x`. This is as described in >>> from sympy import ff, Poly, Symbol >>> from sympy.abc import x >>> n = Symbol('n', integer=True) >>> ff(x, 0) 1 >>> ff(5, 5) 120 >>> ff(x, 5) == x*(x - 1)*(x - 2)*(x - 3)*(x - 4) True >>> ff(Poly(x**2, x), 2) Poly(x**4 - 2*x**3 + x**2, x, domain='ZZ') >>> ff(n, n) factorial(n) Rewriting is complicated unless the relationship between the arguments is known, but falling factorial can be rewritten in terms of gamma, factorial and binomial and rising factorial. >>> from sympy import factorial, rf, gamma, binomial, Symbol >>> n = Symbol('n', integer=True, positive=True) >>> F = ff(n, n - 2) >>> for i in (rf, ff, factorial, binomial, gamma): ... F.rewrite(i) ... RisingFactorial(3, n - 2) FallingFactorial(n, n - 2) factorial(n)/2 binomial(n, n - 2)*factorial(n - 2) gamma(n + 1)/2 See Also ======== factorial, factorial2, RisingFactorial References ========== .. [1] https://mathworld.wolfram.com/FallingFactorial.html .. [2] Peter Paule, "Greatest Factorial Factorization and Symbolic Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268, 1995. c tt|}tjus|tjur tjS|jr|krt S|jr|jr tjS|jrtj ur tj Stj ur|j r tj Stj SttrWj}t|dkrt!dt#fdt%t'|dSt#fdt%t'|dStj ur tj Stj ur tj Sttrkj}t|dkrt!ddt#fdt%dt)t'|dzdz Sdt#fdt%dt)t'|dzdz SdS)Nr,z0ff only defined for polynomials on one generatorc6|| zSrrrs r#rz'FallingFactorial.eval..s./!or%c||z zSrr*rs r#rz'FallingFactorial.eval..rr%rc4||zSrrrs r#rz'FallingFactorial.eval..s011771::r%c||zzSrr*rs r#rz'FallingFactorial.eval.. sAEr%)rrrrqrrXrUrVrrWrrrrrrrrr[rArsrs ` r#rczFallingFactorial.evals AJJ AJJ ::ae5L \, Ja1ffQ<<  \* Jy) Ju =&JAJ z)a0008.#$#55#$:-%a.. <#$6D"4yy1}}&02K'L'L!L(./>/>/>/>.3CFFmmQ(@(@!@$**@*@*@*@*/A--$<$<<AJ z)a000 z)%a.. 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RV'?'W'WWX||E""**;T*BBBr%ct|jdj|jdj|jdjfSrrrs r#rz!FallingFactorial._eval_is_integer/r r%rr) r&r'r(r)rrcrrrrrrr*r%r#rrs<<|2J2J[2Jh ? ? ? ?   111QQQCCCC88888r%rceZdZdZddZedZedZdZdZ dZ dd Z dd Z d Z dZdZdZd S)ra Implementation of the binomial coefficient. It can be defined in two ways depending on its desired interpretation: .. math:: \binom{n}{k} = \frac{n!}{k!(n-k)!}\ \text{or}\ \binom{n}{k} = \frac{(n)_k}{k!} First, in a strict combinatorial sense it defines the number of ways we can choose `k` elements from a set of `n` elements. In this case both arguments are nonnegative integers and binomial is computed using an efficient algorithm based on prime factorization. The other definition is generalization for arbitrary `n`, however `k` must also be nonnegative. This case is very useful when evaluating summations. For the sake of convenience, for negative integer `k` this function will return zero no matter the other argument. To expand the binomial when `n` is a symbol, use either ``expand_func()`` or ``expand(func=True)``. The former will keep the polynomial in factored form while the latter will expand the polynomial itself. See examples for details. Examples ======== >>> from sympy import Symbol, Rational, binomial, expand_func >>> n = Symbol('n', integer=True, positive=True) >>> binomial(15, 8) 6435 >>> binomial(n, -1) 0 Rows of Pascal's triangle can be generated with the binomial function: >>> for N in range(8): ... print([binomial(N, i) for i in range(N + 1)]) ... [1] [1, 1] [1, 2, 1] [1, 3, 3, 1] [1, 4, 6, 4, 1] [1, 5, 10, 10, 5, 1] [1, 6, 15, 20, 15, 6, 1] [1, 7, 21, 35, 35, 21, 7, 1] As can a given diagonal, e.g. the 4th diagonal: >>> N = -4 >>> [binomial(N, i) for i in range(1 - N)] [1, -4, 10, -20, 35] >>> binomial(Rational(5, 4), 3) -5/128 >>> binomial(Rational(-5, 4), 3) -195/128 >>> binomial(n, 3) binomial(n, 3) >>> binomial(n, 3).expand(func=True) n**3/6 - n**2/2 + n/3 >>> expand_func(binomial(n, 3)) n*(n - 2)*(n - 1)/6 In many cases, we can also compute binomial coefficients modulo a prime p quickly using Lucas' Theorem [2]_, though we need to include `evaluate=False` to postpone evaluation: >>> from sympy import Mod >>> Mod(binomial(156675, 4433, evaluate=False), 10**5 + 3) 28625 Using a generalisation of Lucas's Theorem given by Granville [3]_, we can extend this to arbitrary n: >>> Mod(binomial(10**18, 10**12, evaluate=False), (10**5 + 3)**2) 3744312326 References ========== .. [1] https://www.johndcook.com/blog/binomial_coefficients/ .. [2] https://en.wikipedia.org/wiki/Lucas%27s_theorem .. 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