EdmQddlmZmZmZmZddlmZddlmZddl m Z ddl m Z ddl mZddlmZmZddlmZmZdd lmZdd lmZGd d eZGd deZddZdZdZdZdZdS))BasicDictsympifyTuple)Integerdefault_sort_key)_sympifybell)zeros) FiniteSetUnion)flattengroup)as_int) defaultdictceZdZdZdZdZdZd dZedZ dZ dZ dZ d Z ed Zed Zed ZdS) 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<g}d}|D]n}t|tr5t|}t|t|krd}n%|}|t |ot d|Dstdt|}|s*t|td|Dkrtdtj |g|R}t||_ t||_|S)aW 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}) FTc3@K|]}t|tVdSN) isinstancer).0parts >lib/python3.11/site-packages/sympy/combinatorics/partitions.py z$Partition.__new__..Ns,@@4:dI..@@@@@@z@Each argument to Partition should be a list, set, or a FiniteSetc34K|]}t|VdSr)len)rargs rrz$Partition.__new__..Us(99SC999999rz'Partition contained duplicate elements.)rlistsetr appendr all ValueErrorrsumr__new__tuplememberssize)cls partitionargsdupsr!as_setUobjs rr(zPartition.__new__s8H ' 'C#t$$ Sv;;S)DE KK & & & &@@4@@@@@ /.// / 4L  H3q66C99D999999 HFGG G+d+++Ahh q66 rc|j}n&tt|jfd}ttt|j||jfS)aReturn 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})] Nc$t|Srr)worders rz$Partition.sort_key..us+;Au+E+Erkey)r*r)sortedmapr r+rank)selfr6r*s ` rsort_keyzPartition.sort_key]sp(  HlGGF4>> from sympy.combinatorics import Partition >>> Partition([1], [2, 3]).partition [[1], [2, 3]] Nc:g|]}t|tS)r8)r:r rps r z'Partition.partition..s6&:&:&:*+'-Q4D&E&E&E&:&:&:r) _partitionr:r.r=s rr-zPartition.partitionxsE ? ;$&:&:/3y&:&:&:;;DOrct|}|j|z}t|t|jz|j}t ||jS)ai Return permutation whose rank is ``other`` greater than current rank, (mod the maximum rank for the set). Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3]) >>> a.rank 1 >>> (a + 1).rank 2 >>> (a + 100).rank 1 )rr< RGS_unrankRGS_enumr+rfrom_rgsr*)r=otheroffsetresults r__add__zPartition.__add__s]"u U"V$TY//0 I''!!&$,777rc.|| S)af Return permutation whose rank is ``other`` less than current rank, (mod the maximum rank for the set). Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3]) >>> a.rank 1 >>> (a - 1).rank 0 >>> (a - 100).rank 1 )rMr=rJs r__sub__zPartition.__sub__s"||UF###rcp|t|kS)a Checks if a partition is less than or equal to the other based on rank. Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3, 4, 5]) >>> b = Partition([1], [2, 3], [4], [5]) >>> a.rank, b.rank (9, 34) >>> a <= a True >>> a <= b True r>rrOs r__le__zPartition.__le__s)$}}'%.."9"9";";;;rcp|t|kS)aA 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 rRrOs r__lt__zPartition.__lt__s)}}!8!8!:!:::rc^|j|jSt|j|_|jS)z Gets the rank of a partition. Examples ======== >>> from sympy.combinatorics import Partition >>> a = Partition([1, 2], [3], [4, 5]) >>> a.rank 13 )_rankRGS_rankRGSrEs rr<zPartition.ranks/ : : dh'' zrci|j}t|D]\}}|D]}||<tfdtd|DtDS)a 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) c g|] }| Sr\)rirgss rrCz!Partition.RGS.. s/FFFc!fFFFrcg|] }|D]}| Sr\r\)rrBr]s rrCz!Partition.RGS.. s% - - -11 - -aQ - - - -rr8)r- enumerater)r:r )r=r-r]rjr^s @rrYz Partition.RGSs6N  ++  GAt  A FFFFf - - - - -3C'E'E'EFFFGG Grcrt|t|krtdt|dz}dt|D}d}|D](}|||||dz })t d|Dstdt |S)aB 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 lengthscg|]}gSr\r\rr]s rrCz&Partition.from_rgs..*s111AR111rrc3K|]}|VdSrr\rAs rrz%Partition.from_rgs../s"((1((((((rz(some blocks of the partition were empty.)r r&maxranger$r%r)r=r^elementsmax_elemr-rar]s rrIzPartition.from_rgs s4 s88s8}} $ DBCC Cs88a<11x111   A aL   , , , FAA((i((((( IGHH H)$$rr)__name__ __module__ __qualname____doc__rWrDr(r>propertyr-rMrPrSrUr<rY classmethodrIr\rrrrs   EJ<<<|MMMM6  X 8880$$$&<<<(;;;"X" G GX GD#%#%[#%#%#%rrcdeZdZdZdZdZd dZdZdZdZ e dZ dZ d Z dd Zd ZdS)IntegerPartitionaZ 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"|||}}t|ttfrg}tt |dD]?\}}|st |t |}}||g|z@t|}n1tttt |d}d}|t|}d}nt |}|s%t||krtd|ztd|Drtdtj|t|t!|}t ||_||_|S)a Generates a new IntegerPartition object from a list or dictionary. Explantion ========== 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 NTreverseFzPartition did not add to %sc3"K|] }|dkV dS)rcNr\res rrz+IntegerPartition.__new__..s&((q1u((((((rz-All integer summands must be greater than one)rdictrr:r"itemsrextendr)r;r'r&anyrr(rrr-integer)r,r-r{_kvsum_okr2s rr(zIntegerPartition.__new__SsB  4!*GYG i$ . . LAtIOO$5$566EEE  1ayy&))1!QaIIfS%;%;TJJJKKI  &)nnGFFWooG F#i..G3 F:WDEE E ((i((( ( ( NLMM MmC!1!15)3DEEY   rcHtt}|||j}|dgkrt |jdiS|ddkr<||dxxdzcc<|ddkrd|d<n{dx||ddz <|d<ng||dxxdzcc<|d|dz}|d}d|d<|r1|dz}||z dkr!||xx||zz cc<||||zz}|1t |j|S)aReturn 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 rcr)rintupdateas_dict_keysrrr{)r=dkeysleftnews rprev_lexzIntegerPartition.prev_lexs^       z A3; 7#T\1$566 6 8q= ' d2hKKK1 KKKBx1} +!)**$r(Q,!A$$ d2hKKK1 KKKQ4$r(?Dr(CAaD 'q#:?'cFFFdCi'FFFAcF3J&D  '   a000rcbtt}|||j}|d}||jkr ||j|d<n&|dkrb||dkr$||dzxxdz cc<||xxdzcc<n|d}||dzxxdz cc<||dz |z|d<d||<n||dkrlt|dkr-|d||dz<|j|z dz |d<nr|dz}||xxdz cc<|||z|z |d<d||<nF|d}|dz}||xxdz cc<|||z|||zz|z }dx||<||<||d<t|j|S)aReturn 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 rrcrrr) rrrrrr{clearr rr)r=rr9aba1b1needs rnext_lexzIntegerPartition.next_lexs       j G    GGIII>> from sympy.combinatorics.partitions import IntegerPartition >>> IntegerPartition([1]*3 + [2] + [3]*4).as_dict() {1: 3, 2: 1, 3: 4} NF)multiplecg|] }|d S)rr\)rgs rrCz,IntegerPartition.as_dict..s///1!A$///r)_dictrr-rrw)r=groupss rrzIntegerPartition.as_dictsL : &4>E:::F/////DJfDJzrcd}t|jdgz}|d}dg|z}|dkr0|||kr|||dz <|dz}|||k|dz }|dk0|S)a 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] rcr)r"r-)r=ratemp_arrr}rs r conjugatezIntegerPartition.conjugates ''1#- QK CE!e hqk/ !a%Qhqk/  FA !e  rctt|jtt|jkS)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-rOs rrUzIntegerPartition.__lt__s3"HT^,,--Xeo5N5N0O0OOOrctt|jtt|jkS)a 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 rrOs rrSzIntegerPartition.__le__%s3HT^,,--hu6O6O1P1PPPr#cPdfd|jDS)a Prints the ferrer diagram of a partition. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> print(IntegerPartition([1, 1, 5]).as_ferrers()) ##### # #  cg|]}|zSr\r\)rr]chars rrCz/IntegerPartition.as_ferrers..@s999Q$q&999r)joinr-)r=rs `r as_ferrerszIntegerPartition.as_ferrers3s.yy9999$.999:::rcDtt|jSr)strr"r-rEs r__str__zIntegerPartition.__str__Bs4''(((rr)r)rkrlrmrnrrr(rrrrorrUrSrrr\rrrrrr4s6 E E<<<<|#1#1#1J010101d$X.PPP& Q Q Q ; ; ; ;)))))rrrNc^ddlm}t|}|dkrtd||}g}|dkr@|d|}|d||z}|||f|||zz}|dk@|dt d|D}|S)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)_randintrczn must be a positive integerTrtc g|] \}}|g|z Sr\r\)rr}ms rrCz,random_integer_partition..is"55541a!Q555r)sympy.core.randomrrr&r$sortr)nseedrrandintr-r}mults rrandom_integer_partitionrFs(+*****q A1u97888htnnGI q5 GAqMMwq!Q$!T### QtV  q5 NN4N   55955566I rc t|dz}t|dzD] }d|d|f< td|dzD]K}t|D]9}|||z kr'|||dz |fz||dz |dzfz|||f<2d|||f<:L|S)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]]) rcr)r rh)rrr]ras rRGS_generalizedrms& a!e A 1q5\\!Q$ 1a!e__q  AAEz aAqk/Aa!eQUlO;!Q$!Q$   Hrc@|dkrdS|dkrdSt|S)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 rcrr )rs rrHrHs1: Aq q&qAwwrc|dkrtd|dkst||krtddg|dzz}d}t|}td|dzD]J}|||z |f}||z}||kr|dz||<||z}|dz }-t ||z dz||<||z}Kd|ddDS)a 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] rczThe superset size must be >= 1rzInvalid argumentsrcg|]}|dz S)rcr\)rxs rrCzRGS_unrank..s ! ! !aAE ! ! !rN)r&rHrrhr)r<rLraDr]r~crs rrGrGs 1u;9::: ax.8A;;$&.,--- q1u A AA 1a!e__   a!eQhK qS : q5AaD BJD FAAtax!|$$AaD AIDD ! !1QRR5 ! ! !!rct|}d}t|}td|D]L}t||dzd}t|d|}||||dzf||zz }M|S)z Computes the rank of a restricted growth string. Examples ======== >>> from sympy.combinatorics.partitions import RGS_rank, RGS_unrank >>> RGS_rank([0, 1, 2, 1, 3]) 42 >>> RGS_rank(RGS_unrank(4, 7)) 4 rrcN)r rrhrg)r^rgs_sizer<rr]rrs rrXrXs3xxH D!!A 1h  %% QUHH   AaCMM !QU( c!f$$ Krr) sympy.corerrrrsympy.core.numbersrsympy.core.sortingr sympy.core.sympifyr %sympy.functions.combinatorial.numbersr sympy.matricesr sympy.sets.setsrrsympy.utilities.iterablesrrsympy.utilities.miscr collectionsrrrrrrrHrGrXr\rrrs222222222222&&&&&&//////''''''666666 ,,,,,,,,44444444''''''$#####b%b%b%b%b% b%b%b%J O)O)O)O)O)uO)O)O)d$$$$N   @"""J " " "Fr