+ iiQ^RIHtHtHtHt^RIHt^RIHt^RI H t ^RI H t ^RI Ht^RIHtHt^RIHtHt^R IHt^R IHt!R R ]4t!R R]4tRRltRtRtRtRtR#))BasicDictsympifyTuple)Integerdefault_sort_key)_sympifybell)zeros) FiniteSetUnion)flattengroup)as_int) defaultdictca]tRt^toRtRtRtRtRRlt] R4t Rt Rt Rt R t] R 4t] R 4t]R 4tR tVtR#) Partitionz 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 Nc F.pRpVF]p\V\4'd+\V4p\V4\V48dRpM TpVP \ V44K_ \ ;QJdRV4F 'dK RM RM !RV44'g \R4h\V!pV'g!\V4\RV448d \R4h\P!V.VO5!p\V4Vn \V4VnV#)a Generates a new partition object. This method also verifies if the arguments passed are valid and raises a ValueError if they are not. Examples ======== Creating Partition from Python lists: >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3]) >>> a Partition({3}, {1, 2}) >>> a.partition [[1, 2], [3]] >>> len(a) 2 >>> a.members (1, 2, 3) Creating Partition from Python sets: >>> Partition({1, 2, 3}, {4, 5}) Partition({4, 5}, {1, 2, 3}) Creating Partition from SymPy finite sets: >>> from sympy import FiniteSet >>> a = FiniteSet(1, 2, 3) >>> b = FiniteSet(4, 5) >>> Partition(a, b) Partition({4, 5}, {1, 2, 3}) FTc3B"TFp\V\4xK R#5iN) isinstancer).0parts& ~/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/combinatorics/partitions.py $Partition.__new__..Ns@44:dI..4sz@Each argument to Partition should be a list, set, or a FiniteSetc38"TFp\V4xK R#5ir)len)rargs& rrrUs9DSCDsz'Partition contained duplicate elements.)rlistsetr appendr all ValueErrorrsumr__new__tuplememberssize)cls partitionargsdupsr!as_setUobjs&* rr(Partition.__new__sHC#t$$Sv;S)D KK &s@4@sss@4@@@./ / 4L 3q6C9D999FG G+d+Ah q6 c aSfVPpM$\\VPV3RlR74p\\\VP W P 344#)a9Return 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 import default_sort_key >>> from sympy.combinatorics 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 [Partition({1, 2}), Partition({1}, {2}), Partition({1, x}), Partition({3, 4}), Partition({0, 1, 2, 3})] c<\VS4#rr)worders&r$Partition.sort_key..us +;Au+Er4key)r*r)sortedmapr r+rank)selfr8r*s&f rsort_keyPartition.sort_key]sO( =llGF4<>> from sympy.combinatorics import Partition >>> Partition([1], [2, 3]).partition [[1], [2, 3]] r;) _partitionr=r.r )r@ps& rr-Partition.partitionxsP ?? "$/3yy&:/8!'-Q4D&E/8&:;DO&:sAc \V4pVPV,p\V\VP4,VP4p\ P W0P4#)a Return permutation whose rank is ``other`` greater than current rank, (mod the maximum rank for the set). Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3]) >>> a.rank 1 >>> (a + 1).rank 2 >>> (a + 100).rank 1 )rr? RGS_unrankRGS_enumr+rfrom_rgsr*)r@otheroffsetresults&& r__add__Partition.__add__sV"u U"V$TYY/0 II'!!&,,77r4c &VPV)4#)z Return permutation whose rank is ``other`` less than current rank, (mod the maximum rank for the set). Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3]) >>> a.rank 1 >>> (a - 1).rank 0 >>> (a - 100).rank 1 )rNr@rKs&&r__sub__Partition.__sub__s"||UF##r4c VVP4\V4P48*#)a Checks if a partition is less than or equal to the other based on rank. Examples ======== >>> from sympy.combinatorics 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 rArrQs&&r__le__Partition.__le__s"$}}'%."9"9";;;r4c VVP4\V4P48#)z Checks if a partition is less than the other. Examples ======== >>> from sympy.combinatorics 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 rUrQs&&r__lt__Partition.__lt__s"}}!8!8!:::r4c VPe VP#\VP4VnVP#)z Gets the rank of a partition. Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3], [4, 5]) >>> a.rank 13 )_rankRGS_rankRGSr@s&rr?Partition.ranks2 :: !:: dhh' zzr4c  /pVPp\V4Fwr4VFpW1V&K K \\VUUu.uF qfFq3NK K upp\R7Uu.uF q1V,NK up4#uuppiuupi)aD Returns the "restricted growth string" of the partition. Explanation =========== The RGS is returned as a list of indices, L, where L[i] indicates the block in which element i appears. For example, in a partition of 3 elements (a, b, c) into 2 blocks ([c], [a, b]) the RGS is [1, 1, 0]: "a" is in block 1, "b" is in block 1 and "c" is in block 0. Examples ======== >>> from sympy.combinatorics 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 Partition({3}, {4}, {5}, {1, 2}) >>> _.RGS (0, 0, 1, 2, 3) r;)r- enumerater)r=r )r@rgsr-irjrEs& rr^ Partition.RGSs6NN  +GAA,f! - 11aQ1Q -3C'EF'E!ff'EFG G -Fs A9!A?c \V4\V48wd \R4h\V4^,p\V4Uu.uFp.NK pp^pVF)pWT,P W&,4V^, pK+ \ ;QJdRV4F 'dK RM RM !RV44'g \R4h\ V!#uupi)a Creates a set partition from a restricted growth string. Explanation =========== 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 import Partition >>> Partition.from_rgs([0, 1, 2, 0, 1], list('abcde')) Partition({c}, {a, d}, {b, e}) >>> Partition.from_rgs([0, 1, 2, 0, 1], list('cbead')) Partition({e}, {a, c}, {b, d}) >>> a = Partition([1, 4], [2], [3, 5]) >>> Partition.from_rgs(a.RGS, a.members) Partition({2}, {1, 4}, {3, 5}) z#mismatch in rgs and element lengthsc3$"TFqxK R#5ir)rrEs& rr%Partition.from_rgs../s(i1isFTz(some blocks of the partition were empty.)r r&maxranger$r%r)r@rcelementsmax_elemrdr-res&&& rrJPartition.from_rgs s4 s8s8} $BC Cs8a<!&x1AR 1 A L   , FAs(i(sss(i(((GH H)$$2s C)rDr\r)__name__ __module__ __qualname____firstlineno____doc__r\rDr(rApropertyr-rNrRrVrYr?r^ classmethodrJ__static_attributes____classdictcell__ __classdict__s@rrrs  EJ<|M6  80$&<(;"" G GD#%#%r4rcta]tRtRtoRtRtRtRRltRtRt Rt ] R4t R t R tRR ltR tR tVtR#)IntegerPartitioni4a This class represents an integer partition. Explanation =========== In number theory and combinatorics, a partition of a positive integer, ``n``, also called an integer partition, is a way of writing ``n`` as a list of positive integers that sum to n. Two partitions that differ only in the order of summands are considered to be the same partition; if order matters then the partitions are referred to as compositions. For example, 4 has five partitions: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1]; the compositions [1, 2, 1] and [1, 1, 2] are the same as partition [2, 1, 1]. See Also ======== sympy.utilities.iterables.partitions, sympy.utilities.iterables.multiset_partitions References ========== .. [1] https://en.wikipedia.org/wiki/Partition_%28number_theory%29 Nc VeYr\V\\34'dk.p\VP 4RR7F=wrEV'gK\ V4\ V4rTVP V.V,4K? \V4pM$\\\\ V4RR74pRpVf\V4pRpM \ V4pV'g#\V4V8wd\RV,4h\;QJdRV4F 'gK RM RM !RV44'd \R4h\P!V\V4\V!4p\!V4VnW'nV#)aP Generates a new IntegerPartition object from a list or dictionary. Explanation =========== 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 TreverseFzPartition did not add to %sc3*"TF q^8xK R#5i)Nri)rrds& rr+IntegerPartition.__new__..s(i1uisz-All integer summands must be greater than one)rdictrr=itemsrextendr)r>r'r&anyrr(rrr"r-integer)r,r-r_kvsum_okr2s&&& rr(IntegerPartition.__new__Ss(B  !*Y i$ . .Ay0$?ay&)1!Q @ aIfS%;TJKI ?)nGFWoG#i.G3:WDE E 3(i(333(i( ( (LM MmmC!15)3DEY   r4c \\4pVPVP44VPpV^.8Xd\ VP ^/4#VR,^8wdHWR,;;,^,uu&VR,^8Xd^V^&M^;WR,^, &V^&MWR,;;,^,uu&V^,VR,,pVR,p^V^&V'dKV^,pW4, ^8gK W;;,W4,, uu&W1V,V,,pKR\ VP V4#)a:Return the previous partition of the integer, n, in lexical order, wrapping around to [1, ..., 1] if the partition is [n]. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> p = IntegerPartition([4]) >>> print(p.prev_lex()) [3, 1] >>> p.partition > p.prev_lex().partition True )rintupdateas_dict_keysr|r)r@dkeysleftnews& rprev_lexIntegerPartition.prev_lexs    zz A3;#T\\1$56 6 8q= 2hK1 KBx1}!)**r(Q,!A$ 2hK1 KQ4$r(?Dr(CAaDq:?Fdi'FcF3J&D a00r4c \\4pVPVP44VPpVR,pW0P 8Xd"VP 4VP V^&EMtV^8XdW,^8d2W^,;;,^, uu&W;;,^,uu&EM/VR,pW^,;;,^, uu&W,^, V,V^&^W&MW,^8d\V4^8Xd:VP 4^W^,&VP V, ^, V^&MV^,pW;;,^, uu&W,V,V, V^&^W&M\VR,pV^,pW;;,^, uu&W,V,W,V,,V, p^;W&W&Wq^&\VP V4#)a6Return the next partition of the integer, n, in lexical order, wrapping around to [n] if the partition is [1, ..., 1]. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> p = IntegerPartition([3, 1]) >>> print(p.next_lex()) [4] >>> p.partition < p.next_lex().partition True rr) rrrrrrclearr r|)r@rr<aba1b1needs& rnext_lexIntegerPartition.next_lexsw    jj G  GGI<>> from sympy.combinatorics.partitions import IntegerPartition >>> IntegerPartition([1]*3 + [2] + [3]*4).as_dict() {1: 3, 2: 1, 3: 4} F)multiple)_dictrr-rr)r@groupsgs& rrIntegerPartition.as_dictsR :: 4>>E:F(./1A$$/DJfDJzz0sAc ^p\VP4^.,pV^,p^.V,pV^8d/W2V,8dWV^, &V^,pK#V^, pK5V#)z 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] )r"r-)r@retemp_arrrrs& r conjugateIntegerPartition.conjugatesd '1#- QK CE!eqk/!a%Q FAr4c |\\VP44\\VP448#)aReturn 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 r"reversedr-rQs&&rrYIntegerPartition.__lt__s+"HT^^,-Xeoo5N0OOOr4c |\\VP44\\VP448*#)zReturn 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 rrQs&&rrVIntegerPartition.__le__%s+HT^^,-hu6O1PPPr4c nRPVPUu.uF q!V,NK up4#uupi)z Prints the ferrer diagram of a partition. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> print(IntegerPartition([1, 1, 5]).as_ferrers()) ##### # #  )joinr-)r@charrds&& r as_ferrersIntegerPartition.as_ferrers3s-yy$..9.Qq&&.9::9s2c>\\VP44#r)strr"r-r_s&r__str__IntegerPartition.__str__Bs4'((r4)rrr)#)rprqrrrsrtrrr(rrrrurrYrVrrrwrxrys@rr|r|4s\6 E E<|#1J01d$.P& Q ;))r4r|Ncb^RIHp\V4pV^8d \R4hV!V4p.pV^8d<V!^V4pV!^W,4pVP WV34WV,,pKBVP RR7\ VUUu.uFwrWV.V,NK upp4pV#uuppi)a 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] )_randintzn must be a positive integerTr~)sympy.core.randomrrr&r$sortr)nseedrrandintr-rmultms&& rrandom_integer_partitionrFs(+q A1u788tnGI q5 AqMq!$!# tV  NN4N 95941!Q956I 6s B+ c^\V^,4p\V^,4F p^V^V3&K \^V^,4Fcp\V4FQpW0V, 8:d<W1V^, V3,,W^, V^,3,,WV3&KK^WV3&KS Ke V#)a Computes the m + 1 generalized unrestricted growth strings and returns them as rows in matrix. Examples ======== >>> 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]]) )r rl)rrrdres& rRGS_generalizedrms& a!e A 1q5\!Q$1a!e_qAEzAqk/A!eQUlO;Q$Q$  Hr4c<V^8d^#V^8Xd^#\V4#)a% 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 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 r )rs&rrIrIs!: A q&Awr4cV^8d \R4hV^8g\V4V8:d \R4h^.V^,,p^p\V4p\^V^,4FbpWAV, V3,pW6,pWp8:dV^,W%&W,pV^, pK@\ W, ^,4W%&W,pKd VR,Uu.uF q^, NK up#uupi)z 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] zThe superset size must be >= 1zInvalid arguments:rNN)r&rIrrlr) r?rLreDrdrcrxs && rrHrHs 1u9:: ax8A;$&,-- q1u A AA 1a!e_ !eQhK S :q5AD JD FAtx!|$AD IDR5 !5aEE5 !! !sC+c\V4p^p\V4p\^V4FIp\W^,R4p\V^V4pW#WV^,3,W,,, pKK V#)z 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 N)r rrlrk)rcrgs_sizer?rrdrrs& rr]r]sn3xH D!A 1h  UH  AaM !U( cf$$  Kr4r) sympy.corerrrrsympy.core.numbersrsympy.core.sortingr sympy.core.sympifyr %sympy.functions.combinatorial.numbersr sympy.matricesr sympy.sets.setsrrsympy.utilities.iterablesrrsympy.utilities.miscr collectionsrrr|rrrIrHr]rir4rrsc22&/'6 ,4'$b% b%J O)uO)d$N @"J "Fr4