ó ~9­\c@ s ddlmZmZddlmZmZmZddlmZm Z m Z ddl m Z ddl mZddlmZddlmZmZmZddlmZd efd „ƒYZd efd „ƒYZed „Zd„Zd„Zd„Zd„ZdS(iÿÿÿÿ(tprint_functiontdivision(tBasictDicttsympify(tas_inttdefault_sort_keytrange(tbell(tzeros(t FiniteSet(thas_dupstflattentgroup(t defaultdictt PartitioncB seZdZd Zd Zd„Zd d„Zed„ƒZ d„Z d„Z d„Z d„Z ed„ƒZed „ƒZed „ƒZRS( sù This class represents an abstract partition. A partition is a set of disjoint sets whose union equals a given set. See Also ======== sympy.utilities.iterables.partitions, sympy.utilities.iterables.multiset_partitions cG s®|}td„|Dƒƒs+tdƒ‚ntt|gƒdtƒ}t|ƒratdƒ‚ntj|g|D]}t|Œ^qqŒ}t|ƒ|_ t |ƒ|_ |S(sÍ Generates a new partition object. This method also verifies if the arguments passed are valid and raises a ValueError if they are not. Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> a = Partition([1, 2], [3]) >>> a {{3}, {1, 2}} >>> a.partition [[1, 2], [3]] >>> len(a) 2 >>> a.members (1, 2, 3) cs s$|]}t|ttfƒVqdS(N(t isinstancetlistR (t.0tpart((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pys 4ss:Each argument to Partition should be a list or a FiniteSettkeys(Partition contained duplicated elements.( tallt ValueErrortsortedtsumRR R t__new__ttupletmemberstlentsize(tclst partitiontargstxtobj((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyRs  +c s^ˆdkr|j}n$tt|jd‡fd†ƒƒ}ttt|j||jfƒƒS(s£Return a canonical key that can be used for sorting. Ordering is based on the size and sorted elements of the partition and ties are broken with the rank. Examples ======== >>> from sympy.utilities.iterables import default_sort_key >>> from sympy.combinatorics.partitions import Partition >>> from sympy.abc import x >>> a = Partition([1, 2]) >>> b = Partition([3, 4]) >>> c = Partition([1, x]) >>> d = Partition(list(range(4))) >>> l = [d, b, a + 1, a, c] >>> l.sort(key=default_sort_key); l [{{1, 2}}, {{1}, {2}}, {{1, x}}, {{3, 4}}, {{0, 1, 2, 3}}] Rc s t|ˆƒS(N(R(tw(torder(s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pytZtN( tNoneRRRRtmapRRtrank(tselfR$R((R$s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pytsort_keyBs   cC sJ|jdkrCtg|jD]}t|dtƒ^qƒ|_n|jS(sÜReturn partition as a sorted list of lists. Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> Partition([1], [2, 3]).partition [[1], [2, 3]] RN(t _partitionR'RR R(R*tp((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyR]s .cC sKt|ƒ}|j|}t|t|jƒ|jƒ}tj||jƒS(st Return permutation whose rank is ``other`` greater than current rank, (mod the maximum rank for the set). Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> a = Partition([1, 2], [3]) >>> a.rank 1 >>> (a + 1).rank 2 >>> (a + 100).rank 1 (RR)t RGS_unranktRGS_enumRRtfrom_rgsR(R*tothertoffsettresult((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyt__add__ms     cC s|j| ƒS(sq Return permutation whose rank is ``other`` less than current rank, (mod the maximum rank for the set). Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> a = Partition([1, 2], [3]) >>> a.rank 1 >>> (a - 1).rank 0 >>> (a - 100).rank 1 (R4(R*R1((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyt__sub__…scC s|jƒt|ƒjƒkS(s Checks if a partition is less than or equal to the other based on rank. Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> a = Partition([1, 2], [3, 4, 5]) >>> b = Partition([1], [2, 3], [4], [5]) >>> a.rank, b.rank (9, 34) >>> a <= a True >>> a <= b True (R+R(R*R1((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyt__le__˜scC s|jƒt|ƒjƒkS(sL Checks if a partition is less than the other. Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> a = Partition([1, 2], [3, 4, 5]) >>> b = Partition([1], [2, 3], [4], [5]) >>> a.rank, b.rank (9, 34) >>> a < b True (R+R(R*R1((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyt__lt__¬scC s/|jdk r|jSt|jƒ|_|jS(sá Gets the rank of a partition. Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> a = Partition([1, 2], [3], [4, 5]) >>> a.rank 13 N(t_rankR'tRGS_ranktRGS(R*((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyR)½s cC s”i}|j}x5t|ƒD]'\}}x|D]}|||>> from sympy.combinatorics.partitions import Partition >>> a = Partition([1, 2], [3], [4, 5]) >>> a.members (1, 2, 3, 4, 5) >>> a.RGS (0, 0, 1, 2, 2) >>> a + 1 {{3}, {4}, {5}, {1, 2}} >>> _.RGS (0, 0, 1, 2, 3) R(Rt enumerateRRR(R*trgsRtiRtjR-((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyR:Ïs   cC s»t|ƒt|ƒkr'tdƒ‚nt|ƒd}gt|ƒD] }g^qD}d}x-|D]%}||j||ƒ|d7}qcWtd„|Dƒƒs±tdƒ‚nt|ŒS(s  Creates a set partition from a restricted growth string. The indices given in rgs are assumed to be the index of the element as given in elements *as provided* (the elements are not sorted by this routine). Block numbering starts from 0. If any block was not referenced in ``rgs`` an error will be raised. Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> Partition.from_rgs([0, 1, 2, 0, 1], list('abcde')) {{c}, {a, d}, {b, e}} >>> Partition.from_rgs([0, 1, 2, 0, 1], list('cbead')) {{e}, {a, c}, {b, d}} >>> a = Partition([1, 4], [2], [3, 5]) >>> Partition.from_rgs(a.RGS, a.members) {{2}, {1, 4}, {3, 5}} s#mismatch in rgs and element lengthsiics s|] }|VqdS(N((RR-((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pys ss(some blocks of the partition were empty.(RRtmaxRtappendRR(R*R<telementstmax_elemR=RR>((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyR0ïs N(t__name__t __module__t__doc__R'R8R,RR+tpropertyRR4R5R6R7R)R:t classmethodR0(((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyR s  %      tIntegerPartitioncB sweZdZd Zd Zd d„Zd„Zd„Zd„Z e d„ƒZ d„Z d„Z dd „Zd „ZRS( s This class represents an integer partition. In number theory and combinatorics, a partition of a positive integer, ``n``, also called an integer partition, is a way of writing ``n`` as a list of positive integers that sum to n. Two partitions that differ only in the order of summands are considered to be the same partition; if order matters then the partitions are referred to as compositions. For example, 4 has five partitions: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1]; the compositions [1, 2, 1] and [1, 1, 2] are the same as partition [2, 1, 1]. See Also ======== sympy.utilities.iterables.partitions, sympy.utilities.iterables.multiset_partitions Reference: https://en.wikipedia.org/wiki/Partition_%28number_theory%29 cC s„|dk r||}}nt|ttfƒr®g}xett|jƒƒdtƒD]E\}}|snqVnt|ƒt|ƒ}}|j |g|ƒqVWt |ƒ}n!t tt t|ƒdtƒƒ}t }|dkröt |ƒ}t}n t|ƒ}| r.t |ƒ|kr.td|ƒ‚ntd„|DƒƒrStdƒ‚ntj|||ƒ}t|ƒ|_||_|S(sç Generates a new IntegerPartition object from a list or dictionary. The partition can be given as a list of positive integers or a dictionary of (integer, multiplicity) items. If the partition is preceded by an integer an error will be raised if the partition does not sum to that given integer. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([5, 4, 3, 1, 1]) >>> a IntegerPartition(14, (5, 4, 3, 1, 1)) >>> print(a) [5, 4, 3, 1, 1] >>> IntegerPartition({1:3, 2:1}) IntegerPartition(5, (2, 1, 1, 1)) If the value that the partition should sum to is given first, a check will be made to see n error will be raised if there is a discrepancy: >>> IntegerPartition(10, [5, 4, 3, 1]) Traceback (most recent call last): ... ValueError: The partition is not valid treversesPartition did not add to %scs s|]}|dkVqdS(iN((RR=((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pys _ss"The summands must all be positive.N(R'RtdictRRRtitemstTrueRtextendRR(tFalseRRtanyRRRtinteger(RRRPt_tktvtsum_okR"((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyR,s0 +!     cC sCttƒ}|j|jƒƒ|j}|dgkrKtid|j6ƒS|ddkr©||dcd8<|ddkrŒd|d>> from sympy.combinatorics.partitions import IntegerPartition >>> p = IntegerPartition([4]) >>> print(p.prev_lex()) [3, 1] >>> p.partition > p.prev_lex().partition True iiÿÿÿÿiiþÿÿÿi(Rtinttupdatetas_dictt_keysRHRP(R*tdtkeystlefttnew((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pytprev_lexgs(       cC sÜttƒ}|j|jƒƒ|j}|d}||jkr[|jƒ|j|d>> from sympy.combinatorics.partitions import IntegerPartition >>> p = IntegerPartition([3, 1]) >>> print(p.next_lex()) [4] >>> p.partition < p.next_lex().partition True iÿÿÿÿiiiþÿÿÿi( RRURVRWRXRPtclearRRH(R*RYRtatbta1tb1tneed((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pytnext_lexŒs>             cC s]|jdkrVt|jdtƒ}g|D]}|d^q+|_t|ƒ|_n|jS(s[Return the partition as a dictionary whose keys are the partition integers and the values are the multiplicity of that integer. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> IntegerPartition([1]*3 + [2] + [3]*4).as_dict() {1: 3, 2: 1, 3: 4} tmultipleiN(t_dictR'R RRNRXRJ(R*tgroupstg((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyRW¾s  cC sƒd}t|jƒdg}|d}dg|}xI|dkr~x,|||krp|||d<|d8}qEW|d7}q6W|S(s Computes the conjugate partition of itself. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([6, 3, 3, 2, 1]) >>> a.conjugate [5, 4, 3, 1, 1, 1] ii(RR(R*R>ttemp_arrRRR`((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyt conjugateÐs   cC s(tt|jƒƒtt|jƒƒkS(s€Return True if self is less than other when the partition is listed from smallest to biggest. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([3, 1]) >>> a < a False >>> b = a.next_lex() >>> a < b True >>> a == b False (RtreversedR(R*R1((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyR7èscC s(tt|jƒƒtt|jƒƒkS(s Return True if self is less than other when the partition is listed from smallest to biggest. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([4]) >>> a <= a True (RRkR(R*R1((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyR6ûs t#cC s'djg|jD]}||^qƒS(s Prints the ferrer diagram of a partition. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> print(IntegerPartition([1, 1, 5]).as_ferrers()) ##### # # s (tjoinR(R*tcharR=((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyt as_ferrers s cC stt|jƒƒS(N(tstrRR(R*((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyt__str__sN(RCRDRER'RfRXRR]RdRWRFRjR7R6RoRq(((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyRHs ; % 2    cC sßddlm}t|ƒ}|dkr7tdƒ‚n||ƒ}g}xS|dkrž|d|ƒ}|d||ƒ}|j||fƒ|||8}qLW|jdtƒtg|D]\}}|g|^q¹ƒ}|S(s Generates a random integer partition summing to ``n`` as a list of reverse-sorted integers. Examples ======== >>> from sympy.combinatorics.partitions import random_integer_partition For the following, a seed is given so a known value can be shown; in practice, the seed would not be given. >>> random_integer_partition(100, seed=[1, 1, 12, 1, 2, 1, 85, 1]) [85, 12, 2, 1] >>> random_integer_partition(10, seed=[1, 2, 3, 1, 5, 1]) [5, 3, 1, 1] >>> random_integer_partition(1) [1] iÿÿÿÿ(t_randintisn must be a positive integeriRI(tsympy.utilities.randtestRrRRR@tsortRLR (tntseedRrtrandintRRRtmulttm((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pytrandom_integer_partitions   ,cC sÒt|dƒ}x+td|dƒD]}d|d|f>> from sympy.combinatorics.partitions import RGS_generalized >>> RGS_generalized(6) Matrix([ [ 1, 1, 1, 1, 1, 1, 1], [ 1, 2, 3, 4, 5, 6, 0], [ 2, 5, 10, 17, 26, 0, 0], [ 5, 15, 37, 77, 0, 0, 0], [ 15, 52, 151, 0, 0, 0, 0], [ 52, 203, 0, 0, 0, 0, 0], [203, 0, 0, 0, 0, 0, 0]]) ii(R R(RyRYR=R>((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pytRGS_generalizedCs;cC s.|dkrdS|dkr dSt|ƒSdS(sˆ RGS_enum computes the total number of restricted growth strings possible for a superset of size m. Examples ======== >>> from sympy.combinatorics.partitions import RGS_enum >>> from sympy.combinatorics.partitions import Partition >>> RGS_enum(4) 15 >>> RGS_enum(5) 52 >>> RGS_enum(6) 203 We can check that the enumeration is correct by actually generating the partitions. Here, the 15 partitions of 4 items are generated: >>> a = Partition(list(range(4))) >>> s = set() >>> for i in range(20): ... s.add(a) ... a += 1 ... >>> assert len(s) == 15 iiN(R(Ry((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyR/cs   c C s|dkrtdƒ‚n|dks9t|ƒ|krHtdƒ‚ndg|d}d}t|ƒ}xŒtd|dƒD]w}||||f}||}||krÔ|d||<||8}|d7}qt||dƒ||<||;}qWg|dD]}|d^qS(s Gives the unranked restricted growth string for a given superset size. Examples ======== >>> from sympy.combinatorics.partitions import RGS_unrank >>> RGS_unrank(14, 4) [0, 1, 2, 3] >>> RGS_unrank(0, 4) [0, 0, 0, 0] isThe superset size must be >= 1isInvalid argumentsi(RR/R{RRU( R)RytLR>tDR=RStcrR!((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyR.ˆs"      cC sƒt|ƒ}d}t|ƒ}x^td|ƒD]M}t||dƒ}t|d|!ƒ}||||df||7}q.W|S(sð Computes the rank of a restricted growth string. Examples ======== >>> from sympy.combinatorics.partitions import RGS_rank, RGS_unrank >>> RGS_rank([0, 1, 2, 1, 3]) 42 >>> RGS_rank(RGS_unrank(4, 7)) 4 ii(RR{RR?(R<trgs_sizeR)R}R=RuRy((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyR9«s  $N(t __future__RRt sympy.coreRRRtsympy.core.compatibilityRRRt%sympy.functions.combinatorial.numbersRtsympy.matricesR tsympy.sets.setsR tsympy.utilities.iterablesR R R t collectionsRRRHR'RzR{R/R.R9(((s=lib/python2.7/site-packages/sympy/combinatorics/partitions.pyts ÿÿ ' % #