\hiQSSKJrJrJrJr SSKJr SSKJr SSK J r SSK J r SSK Jr SSKJrJr SSKJrJr SS KJr SS KJr "S S \5r"S S\5rSSjrSrSrSrSrg))BasicDictsympifyTuple)Integerdefault_sort_key)_sympifybell)zeros) FiniteSetUnion)flattengroup)as_int) defaultdictc\rSrSrSrSrSrSrSSjr\ S5r Sr Sr S r S r\ S 5r\ S 5r\S 5rSrg) 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/nSnUH[n[U[5(a)[U5n[U5[U5:aSn O UnUR [ U55 M] [ SU55(d [S5e[U6nU(d[U5[SU55:a [S5e[R"U/UQ76n[U5Ul [U5UlU$)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# UHn[U[5v M g7fN) isinstancer).0parts V/var/www/auris/envauris/lib/python3.13/site-packages/sympy/combinatorics/partitions.py $Partition.__new__..Ns@44:dI..4sz@Each argument to Partition should be a list, set, or a FiniteSetc38# UHn[U5v M g7fr)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 &@4@@@./ / 4L 3q6C9D999FG G+d+Ah q6 c^Tc URnO![[URU4SjS95n[[[UR X R 455$)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>[UT5$rr)worders r$Partition.sort_key..us +;Au+Er5key)r+r*sortedmapr r,rank)selfr9r+s ` rsort_keyPartition.sort_key]sO( =llGF4<>> from sympy.combinatorics import Partition >>> Partition([1], [2, 3]).partition [[1], [2, 3]] r<) _partitionr>r/r )rAps rr.Partition.partitionxsP ?? "$/3yy&:/8!'-Q4D&E/8&:;DO&:sAc[U5nURU-n[U[UR5-UR5n[ R X0R5$)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+)rAotheroffsetresults r__add__Partition.__add__sV"u U"V$TYY/0 II'!!&,,77r5c&URU*5$)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 )rOrArLs r__sub__Partition.__sub__s"||UF##r5cVUR5[U5R5:*$)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 rBrrRs r__le__Partition.__le__s"$}}'%."9"9";;;r5cVUR5[U5R5:$)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 rVrRs r__lt__Partition.__lt__s"}}!8!8!:::r5cURb UR$[UR5UlUR$)z Gets the rank of a partition. Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3], [4, 5]) >>> a.rank 13 )_rankRGS_rankRGSrAs rr@Partition.ranks2 :: !:: dhh' zzr5c 0nURn[U5Hup4UHnX1U'M M [[UVVs/sH ofHo3PM M snn[S9Vs/sHo1UPM sn5$s snnfs snf)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 )rArgsr.irjrFs rr_ Partition.RGSs6NN  +GAA,f! - 11aQ1Q -3C'EF'E!f'EFG G -Fs A5 A;cB[U5[U5:wa [S5e[U5S-n[U5Vs/sHn/PM nnSnUHnXTR X&5 US- nM [ SU55(d [S5e[ U6$s snf)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 lengthsrc3$# UHov M g7fr)rrFs rr%Partition.from_rgs../s(i1isz(some blocks of the partition were empty.)r!r'maxranger%r&r)rArdelementsmax_elemrer.rfs rrKPartition.from_rgs s4 s8s8} $BC Cs8a<!&x1AR 1 A L   , FA(i(((GH H)$$2s B)rEr]r)__name__ __module__ __qualname____firstlineno____doc__r]rEr)rBpropertyr.rOrSrWrZr@r_ classmethodrK__static_attributes__rkr5rrrs  EJ<|M6  80$&<(;"" G GD#%#%r5rch\rSrSrSrSrSrSSjrSrSr Sr \ S5r S r S rSS jrS rS rg)IntegerPartitioni4a This class represents an integer partition. Explanation =========== In number theory and combinatorics, a partition of a positive integer, ``n``, also called an integer partition, is a way of writing ``n`` as a list of positive integers that sum to n. Two partitions that differ only in the order of summands are considered to be the same partition; if order matters then the partitions are referred to as compositions. For example, 4 has five partitions: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1]; the compositions [1, 2, 1] and [1, 1, 2] are the same as partition [2, 1, 1]. See Also ======== sympy.utilities.iterables.partitions, sympy.utilities.iterables.multiset_partitions References ========== .. [1] https://en.wikipedia.org/wiki/Partition_%28number_theory%29 NcUbXp[U[[45(ab/n[UR 5SS9H8upEU(dM[ U5[ U5pTUR U/U-5 M: [U5nO![[[[ U5SS95nSnUc[U5nSnO [ U5nU(d[U5U:wa[SU-5e[SU55(a [S5e[R"U[U5[U65n[!U5UlX'lU$)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*# UH oS:v M g7f)riNrk)rres 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_okr3s rr)IntegerPartition.__new__SsB  !*Y i$ . .Ay0$?ay&)1!Q @ aIfS%;TJKI ?)nGFWoG#i.G3:WDE E (i( ( (LM MmmC!15)3DEY   r5c[[5nURUR55 URnUS/:Xa[ UR S05$USS:wa.XS==S-ss'USS:XaSUS'OiS=XSS- 'US'OYXS==S-ss'USUS-nUSnSUS'U(a.US-nX4- S:aX==X4-- ss'X1UU--nU(aM.[ UR U5$)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 rir)rintupdateas_dict_keysr{r)rAdkeysleftnews rprev_lexIntegerPartition.prev_lexs     zz A3;#T\\1$56 6 8q= 2hK1 KBx1}!)**r(Q,!A$ 2hK1 KQ4$r(?Dr(CAaDq:?Fdi'FcF3J&D $   a00r5c[[5nURUR55 URnUSnX0R :Xa UR 5 UR US'OUS:XaJXS:aXS-==S- ss'X==S-ss'OUSnXS-==S- ss'XS- U-US'SX'OXS:a_[U5S:Xa-UR 5 SXS-'UR U- S- US'OWUS-nX==S- ss'XU-U- US'SX'O4USnUS-nX==S- ss'XU-XU--U- nS=X'X'XqS'[UR U5$)a6Return the next partition of the integer, n, in lexical order, wrapping around to [n] if the partition is [1, ..., 1]. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> p = IntegerPartition([3, 1]) >>> print(p.next_lex()) [4] >>> p.partition < p.next_lex().partition True rrirrr) rrrrrrclearr!r{)rArr=aba1b1needs rnext_lexIntegerPartition.next_lexsw    jj G  GGI<[URSS9nUVs/sHo"SPM snUl[ U5UlUR$s snf)aReturn 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} F)multipler)_dictrr.rr)rAgroupsgs rrIntegerPartition.as_dictsP :: 4>>E:F(./1A$/DJfDJzz0sAcSn[UR5S/-nUSnS/U-nUS:a+X2U:aXUS- 'US-nX2U:aMUS- nUS:aM+U$)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] rir)r#r.)rArftemp_arrrrs r conjugateIntegerPartition.conjugates| '1#- QK CE!eqk/!a%Qqk/ FA !e r5c|[[UR55[[UR55:$)aReturn True if self is less than other when the partition is listed from smallest to biggest. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([3, 1]) >>> a < a False >>> b = a.next_lex() >>> a < b True >>> a == b False r#reversedr.rRs rrZIntegerPartition.__lt__s+"HT^^,-Xeoo5N0OOOr5c|[[UR55[[UR55:*$)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 rrRs rrWIntegerPartition.__le__%s+HT^^,-hu6O1PPPr5chSRURVs/sHo!U-PM sn5$s snf)z Prints the ferrer diagram of a partition. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> print(IntegerPartition([1, 1, 5]).as_ferrers()) ##### # #  )joinr.)rAcharres r as_ferrersIntegerPartition.as_ferrers3s+yy$..9.Qq&.9::9s/c>[[UR55$r)strr#r.r`s r__str__IntegerPartition.__str__Bs4'((r5)rrr)#)rrrsrtrurvrrr)rrrrwrrZrWrrryrkr5rr{r{4sT6 E E<|#1J01d$.P& Q ;)r5r{NcFSSKJn [U5nUS:a [S5eU"U5n/nUS:a5U"SU5nU"SX-5nUR XV45 XU--nUS:aM5UR SS9 [ UVVs/sH upWU/U-PM snn5nU$s snnf)a Generates a random integer partition summing to ``n`` as a list of reverse-sorted integers. Examples ======== >>> from sympy.combinatorics.partitions import random_integer_partition For the following, a seed is given so a known value can be shown; in practice, the seed would not be given. >>> random_integer_partition(100, seed=[1, 1, 12, 1, 2, 1, 85, 1]) [85, 12, 2, 1] >>> random_integer_partition(10, seed=[1, 2, 3, 1, 5, 1]) [5, 3, 1, 1] >>> random_integer_partition(1) [1] r)_randintrizn must be a positive integerTr})sympy.core.randomrrr'r%sortr)nseedrrandintr.rmultms rrandom_integer_partitionrFs(+q A1u788tnGI q5 AqMq!$!# tV  q5 NN4N 95941!Q956I 6sB c[US-5n[US-5H nSUSU4'M [SUS-5HBn[U5H0nX0U- ::aX1US- U4-XS- US-4-XU4'M*SXU4'M2 MD U$)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]]) rir)r rn)rrrerfs rRGS_generalizedrms& a!e A 1q5\!Q$1a!e_qAEzAqk/A!eQUlO;Q$Q$  Hr5c4US:agUS:Xag[U5$)a% RGS_enum computes the total number of restricted growth strings possible for a superset of size m. Examples ======== >>> from sympy.combinatorics.partitions import RGS_enum >>> from sympy.combinatorics import Partition >>> RGS_enum(4) 15 >>> RGS_enum(5) 52 >>> RGS_enum(6) 203 We can check that the enumeration is correct by actually generating the partitions. Here, the 15 partitions of 4 items are generated: >>> a = Partition(list(range(4))) >>> s = set() >>> for i in range(20): ... s.add(a) ... a += 1 ... >>> assert len(s) == 15 rirr )rs rrJrJs!: A q&Awr5cjUS:a [S5eUS:d[U5U::a [S5eS/US--nSn[U5n[SUS-5H=nXAU- U4nX6-nXp::aUS-X%'X-nUS- nM'[ X- S-5X%'X-nM? USSVs/sHoS- PM sn$s snf)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] rizThe superset size must be >= 1rzInvalid argumentsrN)r'rJrrnr) r@rLrfDrercrxs rrIrIs 1u9:: ax8A;$&,-- q1u A AA 1a!e_ !eQhK S :q5AD JD FAtx!|$AD IDQR5 !5aE5 !! !sB0c[U5nSn[U5n[SU5H1n[XS-S5n[USU5nX#XVS-4X-- nM3 U$)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 rriN)r!rrnrm)rdrgs_sizer@rrerrs rr^r^sn3xH D!A 1h  UH  AaM !U( cf$$  Kr5r) sympy.corerrrrsympy.core.numbersrsympy.core.sortingr sympy.core.sympifyr %sympy.functions.combinatorial.numbersr sympy.matricesr sympy.sets.setsrrsympy.utilities.iterablesrrsympy.utilities.miscr collectionsrrr{rrrJrIr^rkr5rrsc22&/'6 ,4'$b% b%J O)uO)d$N @"J "Fr5