o 8VaQ@sddlmZmZmZddlmZmZddlmZddl m Z ddl m Z ddl mZmZddlmZmZddlmZGd d d eZGd d d eZdddZddZddZddZddZd S))BasicDictsympify)as_intdefault_sort_key)_sympifybell)zeros) FiniteSetUnion)flattengroup) defaultdictc@szeZdZdZdZdZddZdddZeddZ d d Z d d Z d dZ ddZ eddZeddZeddZdS) 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 NcGsg}d}|D] }t|trt|}t|t|krd}n |}|t|qtdd|Ds4tdt|}|sGt|t dd|DkrKtdt j |g|R}t ||_ t||_|S)al 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.partitions 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.sets.sets import FiniteSet >>> a = FiniteSet(1, 2, 3) >>> b = FiniteSet(4, 5) >>> Partition(a, b) Partition({4, 5}, {1, 2, 3}) FTcss|]}t|tVqdSN) isinstancer ).0partr@/usr/lib/python3/dist-packages/sympy/combinatorics/partitions.py Ksz$Partition.__new__..z@Each argument to Partition should be a list, set, or a FiniteSetcss|]}t|VqdSr)len)rargrrrrRz'Partition contained duplicate elements.)rlistsetrappendrall ValueErrorr sumr __new__tuplememberssize)cls partitionargsZdupsrZas_setUobjrrrr!s*$   zPartition.__new__csBdur|j}n tt|jfddd}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.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 [Partition({1, 2}), Partition({1}, {2}), Partition({1, x}), Partition({3, 4}), Partition({0, 1, 2, 3})] Ncs t|Sr)r)worderrrrs z$Partition.sort_key..key)r#r"sortedmaprr$rank)selfr,r#rr+rsort_keyZs  zPartition.sort_keycCs&|jdurtdd|jD|_|jS)zReturn partition as a sorted list of lists. Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> Partition([1], [2, 3]).partition [[1], [2, 3]] NcSsg|]}t|tdqS)r.)r0rrprrr sz'Partition.partition..) _partitionr0r'r3rrrr&us  zPartition.partitioncCs6t|}|j|}t|t|j|j}t||jS)at 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 )rr2 RGS_unrankRGS_enumr$rfrom_rgsr#)r3otheroffsetresultrrr__add__s zPartition.__add__cCs || S)aq 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 )r@r3r=rrr__sub__s zPartition.__sub__cCs|t|kS)a 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 r4rrArrr__le__szPartition.__le__cCs|t|kS)aL 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 rCrArrr__lt__szPartition.__lt__cCs"|jdur|jSt|j|_|jS)z Gets the rank of a partition. Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> a = Partition([1, 2], [3], [4, 5]) >>> a.rank 13 N)_rankRGS_rankRGSr9rrrr2s  zPartition.rankcsVi|j}t|D] \}}|D]}||<qq tfddtdd|DtdDS)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.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 Partition({3}, {4}, {5}, {1, 2}) >>> _.RGS (0, 0, 1, 2, 3) csg|]}|qSrrrirgsrrr7z!Partition.RGS..cSsg|] }|D]}|qqSrr)rr6rJrrrr7r.)r& enumerater"r0r)r3r&rJrjrrKrrHs  z Partition.RGScCst|t|kr tdt|d}ddt|D}d}|D]}|||||d7}qtdd|Ds>> from sympy.combinatorics.partitions 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 lengthscSsg|]}gqSrrrIrrrr7'sz&Partition.from_rgs..rcss|]}|VqdSrrr5rrrr,sz%Partition.from_rgs..z(some blocks of the partition were empty.)rrmaxrangerrr)r3rLelementsZmax_elemr&rPrJrrrr< s  zPartition.from_rgsr)__name__ __module__ __qualname____doc__rFr8r!r4propertyr&r@rBrDrEr2rH classmethodr<rrrrr s$  >   "rc@sheZdZdZdZdZdddZddZddZd d Z e d d Z d dZ ddZ dddZddZdS)IntegerPartitionaS 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 ========== https://en.wikipedia.org/wiki/Partition_%28number_theory%29 NcCs|dur ||}}t|ttfr:g}tt|ddD]\}}|s#qt|t|}}||g|qt|}n ttt t|dd}d}|durRt |}d}nt|}|sdt ||krdt d|t dd|Drqt dt |||}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 %scss|]}|dkVqdS)rQNrrIrrrrrz+IntegerPartition.__new__..z-All integer summands must be greater than one)rdictrr0ritemsrextendr"r1r ranyrr!r&integer)r%r&rb_kvZsum_okr)rrrr!Ps0!    zIntegerPartition.__new__cCstt}|||j}|dgkrt|jdiS|ddkrB||dd8<|ddkr5d|d<nGd||dd<|d<n:||dd8<|d|d}|d}d|d<|r||d8}||dkrz||||7<||||8}|s^t|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 rQr)rintupdateas_dict_keysr[rb)r3dkeysleftnewrrrprev_lexs*      zIntegerPartition.prev_lexcCstt}|||j}|d}||jkr!||j|d<n|dkr[||dkr>||dd7<||d8<n|d}||dd7<||d||d<d||<ne||dkrt|dkr{|d||d<|j|d|d<nE|d}||d7<|||||d<d||<n*|d}|d}||d7<|||||||}d||<||<||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 rfrQrgrhr) rrirjrkrlrbclearrr[)r3rmr/abZa1Zb1Zneedrrrnext_lexs>         zIntegerPartition.next_lexcCs8|jdurt|jdd}dd|D|_t||_|jS)a[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} NF)ZmultiplecSsg|]}|dqS)rr)rgrrrr7rMz,IntegerPartition.as_dict..)_dictrr&rlr^)r3groupsrrrrks  zIntegerPartition.as_dictcCsnd}t|jdg}|d}dg|}|dkr5|||kr-|||d<|d8}|||ks|d7}|dks|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] rQr)rr&)r3rPZtemp_arrrdrtrrr conjugates     zIntegerPartition.conjugatecCstt|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 rreversedr&rArrrrEszIntegerPartition.__lt__cCstt|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 rzrArrrrD"s zIntegerPartition.__le__#csdfdd|jDS)a Prints the ferrer diagram of a partition. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> print(IntegerPartition([1, 1, 5]).as_ferrers()) ##### # #  csg|]}|qSrrrIcharrrr7=rMz/IntegerPartition.as_ferrers..)joinr&)r3rrr~r as_ferrers0s zIntegerPartition.as_ferrerscCstt|jSr)strrr&r9rrr__str__?szIntegerPartition.__str__r)r|)rUrVrWrXrwrlr!rqrurkrYryrErDrrrrrrr[1s >%2   r[NcCsddlm}t|}|dkrtd||}g}|dkr9|d|}|d||}|||f|||8}|dks|jddtdd|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)_randintrQzn must be a positive integerTr\cSsg|] \}}|g|qSrr)rrdmrrrr7frNz,random_integer_partition..)Zsympy.testing.randtestrrrrsortr )nZseedrZrandintr&rdZmultrrrrandom_integer_partitionCs    rcCst|d}td|dD]}d|d|f<q td|dD].}t|D]'}|||krD|||d|f||d|df|||f<q#d|||f<q#q|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]]) rQr)r rS)rrmrJrPrrrRGS_generalizedjs   2rcCs |dkrdS|dkr dSt|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.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 rQrr)rrrrr;s r;cCs|dkrtd|dkst||krtddg|d}d}t|}td|dD]/}||||f}||}||krK|d||<||8}|d7}q*t||d||<||;}q*dd|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] rQzThe superset size must be >= 1rzInvalid argumentsrgcSsg|]}|dqS)rQr)rxrrrr7rMzRGS_unrank..N)rr;rrSri)r2rLrPDrJreZcrrrrr:s"   r:cCsht|}d}t|}td|D]"}t||dd}t|d|}||||df||7}q|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 rrQN)rrrSrR)rLZrgs_sizer2rrJrrrrrrGs rGr)Z sympy.corerrrZsympy.core.compatibilityrrZsympy.core.sympifyrZ%sympy.functions.combinatorial.numbersr Zsympy.matricesr Zsympy.sets.setsr r Zsympy.utilities.iterablesr r collectionsrrr[rrr;r:rGrrrrs&    ' ' % #