B [@s^ddlmZmZddlmZddlmZddZddZe e fdd Z d d Z e fd d Z dS))print_functiondivision) defaultdict)rangecCsld|fd|dfdi}d}xNtd|ddD]8}|||d|}|||||f<||||f<q,W|S)aReturn a dictionary containing pairs :math:`{(k1,k2) : C_kn}` where :math:`C_kn` are binomial coefficients and :math:`n=k1+k2`. Examples ======== >>> from sympy.ntheory import binomial_coefficients >>> binomial_coefficients(9) {(0, 9): 1, (1, 8): 9, (2, 7): 36, (3, 6): 84, (4, 5): 126, (5, 4): 126, (6, 3): 84, (7, 2): 36, (8, 1): 9, (9, 0): 1} See Also ======== binomial_coefficients_list, multinomial_coefficients r)r)ndakr 8lib/python3.7/site-packages/sympy/ntheory/multinomial.pybinomial_coefficientss $rcCsZdg|d}d}xBtd|ddD],}|||d|}|||<|||<q&W|S)aL Return a list of binomial coefficients as rows of the Pascal's triangle. Examples ======== >>> from sympy.ntheory import binomial_coefficients_list >>> binomial_coefficients_list(9) [1, 9, 36, 84, 126, 126, 84, 36, 9, 1] See Also ======== binomial_coefficients, multinomial_coefficients rr)r)rr r r r r r binomial_coefficients_lists rcsxsr iSddiSdkr$tSfddtD}|dfdd|D}fdd Ddi}dgdd}||d<xtdddD]ĉtt}xtdtdD]v} d| } | sq|| } xR|| D]B\} } d d| | D}||| | 7<||s||=qWqWfd d|D}||<||qW|S) aeReturn a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}`` where ``C_kn`` are multinomial coefficients such that ``n=k1+k2+..+km``. For example: >>> from sympy import multinomial_coefficients >>> multinomial_coefficients(2, 5) # indirect doctest {(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1} The algorithm is based on the following result: Consider a polynomial and its ``n``-th exponent:: P(x) = sum_{i=0}^m p_i x^i P(x)^n = sum_{k=0}^{m n} a(n,k) x^k The coefficients ``a(n,k)`` can be computed using the J.C.P. Miller Pure Recurrence [see D.E.Knuth, Seminumerical Algorithms, The art of Computer Programming v.2, Addison Wesley, Reading, 1981;]:: a(n,k) = 1/(k p_0) sum_{i=1}^m p_i ((n+1)i-k) a(n,k-i), where ``a(n,0) = p_0^n``. r rrcs(g|] }d|dd|dqS))r)rrr ).0i)mr r Ysz-multinomial_coefficients0..rcs$g|]}dd|DqS)css|]\}}||VqdS)Nr )raabbr r r [sz7multinomial_coefficients0...r )rs)_tuple_zips0r r r[sc3s|]}|VqdS)Nr )rr)rr r r\sz,multinomial_coefficients0..cSsg|]\}}||qSr r )rrrr r r rgscsg|]\}}||fqSr r )rtc)r r r rks)rritemsrintminupdate)rrrrZsymbolsZp0rlr rZnnrZt2Zc2ZttZr1r )rrr rrrr multinomial_coefficients07s8 r#c Cs||s|r iSddiS|dkr$t|S|d|krF|dkrFtt||S|gdg|d}t|di}|rpd}n|}x||dkrv||}|rd||<||d<|dkr||dd7<d}d}d}n,|d7}|d}|t|}||d7<xNt||D]@}||r||d8<||t|7}||d7<qW|dd8<||||d|t|<qxW|S)aReturn a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}`` where ``C_kn`` are multinomial coefficients such that ``n=k1+k2+..+km``. For example: >>> from sympy.ntheory import multinomial_coefficients >>> multinomial_coefficients(2, 5) # indirect doctest {(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1} The algorithm is based on the following result: .. math:: \binom{n}{k_1, \ldots, k_m} = \frac{k_1 + 1}{n - k_1} \sum_{i=2}^m \binom{n}{k_1 + 1, \ldots, k_i - 1, \ldots} Code contributed to Sage by Yann Laigle-Chapuy, copied with permission of the author. See Also ======== binomial_coefficients_list, binomial_coefficients r rrr)rdict!multinomial_coefficients_iteratortupler) rrrr!jtjstartvr r r r multinomial_coefficientsqsF    r+c csf|d|ks|dkrBt||}x|D]\}}||fVq(Wn t||}i}x&|D]\}}|||td|<qZW|}|gdg|d}||}|td|} ||| fV|rd} n|} x| |dkr`|| } | rd|| <| |d<| dkr|| dd7<d} n| d7} || d7<|dd8<||}|td|} ||| fVqWdS)aqmultinomial coefficient iterator This routine has been optimized for `m` large with respect to `n` by taking advantage of the fact that when the monomial tuples `t` are stripped of zeros, their coefficient is the same as that of the monomial tuples from ``multinomial_coefficients(n, n)``. Therefore, the latter coefficients are precomputed to save memory and time. >>> from sympy.ntheory.multinomial import multinomial_coefficients >>> m53, m33 = multinomial_coefficients(5,3), multinomial_coefficients(3,3) >>> m53[(0,0,0,1,2)] == m53[(0,0,1,0,2)] == m53[(1,0,2,0,0)] == m33[(0,1,2)] True Examples ======== >>> from sympy.ntheory.multinomial import multinomial_coefficients_iterator >>> it = multinomial_coefficients_iterator(20,3) >>> next(it) ((3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), 1) rrNr)r+rfilter) rrrZmcr r*Zmc1rZt1br'r(r r r r%s<   r%N)Z __future__rr collectionsrZsympy.core.compatibilityrrrr&zipr#r+r%r r r r s  :D