U L?h5#@sddlmZmZmZmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZddlmZmZmZddlmZmZddlmZddlmZddlZddlmZd d Zd d Z d dZ!eddge"e"dddZ#e"e"dddZ$edddddddZ%dS))fzerofrom_int from_rationalfonefhalfbitcountto_intmpf_mulmpf_divmpf_submpf_addmpf_sqrtmpf_pi mpf_cosh_sinhmpf_cosmpf_sin)gcdlegendrejacobi)_sqrt_mod_prime_poweris_quad_residue) deprecated)recurrence_memoNcountcCsd}dg|adg|at|dd}td|D]:}t|dkr2t||||D]}t|dkrR|t|<qRq2td|D]h}t|dkr|t|<|dt|<qxt|}||}||dkrt||t|<qxt||dt|<qxdS)N順rr?)_factor_totientintrange)ZmaxnZlimijxyr(K/var/www/html/venv/lib/python3.8/site-packages/sympy/ntheory/partitions_.py_pre s&        r*cCsR|dkr tS|}d}t|}||dkr:||}|d7}q||}dd|}t|}|dkr|dkrd|} | || }|td|d| | }t|d|dd} tttd| ||t| |} tttd |t| d| t t|||t | ||S|dkrd|} | || }|dkrH|td |dd| | }t|d|dd} tttd| ||t| |} tttdd |dt | dt t|d||t | ||S|||}||dkr|dkrtttd|t t|||St St ||st S||d|d} |td | d|}t|||d} tttd| ||t||} tttdtd|t t|||t| ||S|dks|dkrrt|dt|d} }d| |}||||dd| t||||t|d||}| |||dd|t|||| t|d||}tt|||t||||S|dkrd|d td t|d||}d|d|ddd|ddd}tttd t||||t|||Sd|dtdt|d||}d||ddddd}tt|||t||||S)z Compute the inner sum in HRR formula [1]_ References ========== .. [1] https://msp.org/pjm/1956/6-1/pjm-v6-n1-p18-p.pdf rrr @i@r )rr rpowrr r rrr rrrrrrr!_a)nkprecZk1epZk2vpimodmargZ_phiZd1Zd2Zn1Zn2r(r(r)r5$s                    $,  $ r5c Cst|}t|}t|||}tt|tdd||}t||}tt|||\} } tt||tt||||} tt|| t| |||} t| | S)z Compute the sinh term in the outer sum of the HRR formula. The constants sqrt(2/3*pi) and sqrt(8) must be precomputed. rr+)rrr r rr rr ) r6r%r8sq23pisqrt8r<abcchshDEr(r(r)_dzs  rI)r6returncCszd}d}tD]f}|d|d7}||}|dkr6qv||}||d8}d|kr^|||7}||drn| n|7}q|S)zt Calculate the partition function P(n) Parameters ========== n : int nonnegative integer rr.rrr)r6prevr;Zpentar$npsr(r(r)_partition_recs    rNcsB|dkr dS|dkr$|tdks8tdkr@|dkr@t|SdtkrPtttjd|ddtd |td d td }t|d d }}dtjddt dtjt ddtjt dfdd}t dt |d}|||dkst |dkrB|||dkrB|d}q|}|d}||d kr||d}|||}dkr|}n|dkrN|}qN|} | dkrt dt} tttdd||t||} ttd|} td | D]H} t|| |}t|| || | }t| t|||} ttt|d}qttt| t|S)zX Calculate the partition function P(n) Parameters ========== n : int ri@ Fri@8r g@rr/ rg?d,r.KgUUUUUU?cs<tj}|||||dt|||S)Nr)mathsqrtsinh)r6NrVc1c2c3r(r)_Msz_partition.._Mr-(rz Input too bigr,2)rNZ cache_lengthglobalsr*r"rUr<loglog2rVmaxceilAssertionError ValueErrorrr r rrrr#r5rIr rabsrr)r6Zpbitsr8r:r]bigZsmallrXZerMrMr@rAqrBdr(rYr) _partitionsb         rlzpThe `sympy.ntheory.partitions_.npartitions` has been moved to `sympy.functions.combinatorial.numbers.partition`.z1.13z%deprecated-ntheory-symbolic-functions)Zdeprecated_since_versionZactive_deprecations_targetFcCsddlm}||S)a  Calculate the partition function P(n), i.e. the number of ways that n can be written as a sum of positive integers. .. deprecated:: 1.13 The ``npartitions`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.partition` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. P(n) is computed using the Hardy-Ramanujan-Rademacher formula [1]_. The correctness of this implementation has been tested through $10^{10}$. Examples ======== >>> from sympy.functions.combinatorial.numbers import partition >>> partition(25) 1958 References ========== .. [1] https://mathworld.wolfram.com/PartitionFunctionP.html r) partition)Z%sympy.functions.combinatorial.numbersrm)r6verboseZfunc_partitionr(r(r) npartitionss! ro)F)&Z mpmath.libmprrrrrrrr r r r r rrrrZsympy.external.gmpyrrrZresidue_ntheoryrrZsympy.utilities.decoratorrZsympy.utilities.memoizationrrU itertoolsrr*r5rIr"rNrlror(r(r(r)s"H   V M