B [N@sddlmZmZddlmZmZmZddlmZm Z m Z ddl m Z ddl mZddlmZddlmZmZmZddlmZGd d d eZGd d d eZdddZddZddZddZddZd S))print_functiondivision)BasicDictsympify)as_intdefault_sort_keyrange)bell)zeros) FiniteSet)has_dupsflattengroup) 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 NcGsr|}tdd|Dstdtt|gtd}t|r@tdtj|fdd|D}t||_ t ||_ |S)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 ======== >>> 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) css|]}t|ttfVqdS)N) isinstancelistr ).0partr=lib/python3.7/site-packages/sympy/combinatorics/partitions.py 4sz$Partition.__new__..z:Each argument to Partition should be a list or a FiniteSet)keyz(Partition contained duplicated elements.cSsg|] }t|qSr)r )rxrrr =sz%Partition.__new__..) all ValueErrorsortedsumrr r __new__tuplememberslensize)cls partitionargsobjrrrr s  zPartition.__new__csBdkr|j}ntt|jfddd}ttt|j||jfS)aReturn a canonical key that can be used for sorting. Ordering is based on the size and sorted elements of the partition and ties are broken with the rank. Examples ======== >>> from sympy.utilities.iterables import default_sort_key >>> from sympy.combinatorics.partitions import Partition >>> from sympy.abc import x >>> a = Partition([1, 2]) >>> b = Partition([3, 4]) >>> c = Partition([1, x]) >>> d = Partition(list(range(4))) >>> l = [d, b, a + 1, a, c] >>> l.sort(key=default_sort_key); l [{{1, 2}}, {{1}, {2}}, {{1, x}}, {{3, 4}}, {{0, 1, 2, 3}}] Ncs t|S)N)r)w)orderrrZsz$Partition.sort_key..)r)r"r!rrmaprr$rank)selfr*r"r)r*rsort_keyBs zPartition.sort_keycCs&|jdkr tdd|jD|_|jS)zReturn partition as a sorted list of lists. Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> Partition([1], [2, 3]).partition [[1], [2, 3]] NcSsg|]}t|tdqS))r)rr)rprrrrisz'Partition.partition..) _partitionrr')r.rrrr&]s zPartition.partitioncCs6t|}|j|}t|t|j|j}t||jS)at Return permutation whose rank is ``other`` greater than current rank, (mod the maximum rank for the set). Examples ======== >>> from sympy.combinatorics.partitions import Partition >>> a = Partition([1, 2], [3]) >>> a.rank 1 >>> (a + 1).rank 2 >>> (a + 100).rank 1 )rr- RGS_unrankRGS_enumr$rfrom_rgsr")r.otheroffsetresultrrr__add__ms   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 )r8)r.r5rrr__sub__szPartition.__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 )r/r)r.r5rrr__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 )r/r)r.r5rrr__lt__szPartition.__lt__cCs"|jdk r|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_rankRGS)r.rrrr-s  zPartition.rankcs^i|j}x*t|D]\}}x|D] }||<q"WqWtfddtdd|DtdDS)a Returns the "restricted growth string" of the partition. 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 {{3}, {4}, {5}, {1, 2}} >>> _.RGS (0, 0, 1, 2, 3) csg|] }|qSrr)ri)rgsrrrsz!Partition.RGS..cSsg|]}|D]}|q qSrr)rr0r?rrrrs)r)r& enumerater!rr)r.r&r?rjr)r@rr>s z Partition.RGScCst|t|krtdt|d}ddt|D}d}x&|D]}|||||d7}q@Wtdd|Ds|tdt|S) a  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}} z#mismatch in rgs and element lengthscSsg|]}gqSrr)rr?rrrr sz&Partition.from_rgs..rcss|] }|VqdS)Nr)rr0rrrrsz%Partition.from_rgs..z(some blocks of the partition were empty.)r#rmaxr appendrr)r.r@elementsZmax_elemr&rBr?rrrr4s   zPartition.from_rgs)N)__name__ __module__ __qualname____doc__r<r1r r/propertyr&r8r9r:r;r-r> classmethodr4rrrrr s %    rc@sheZdZdZdZdZdddZddZddZd d Z e d d Z d dZ ddZ dddZddZdS)IntegerPartitiona 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: http://en.wikipedia.org/wiki/Partition_%28number_theory%29 NcCs|dk r||}}t|ttfrxg}xHtt|ddD]0\}}|sHq:t|t|}}||g|q:Wt|}nttt t|dd}d}|dkrt |}d}nt|}|st ||krt d|t dd|Drt dt |||}t||_||_|S) a 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 NT)reverseFzPartition did not add to %scss|]}|dkVqdS)rCNr)rr?rrrr_sz+IntegerPartition.__new__..z"The summands must all be positive.)rdictrrritemsrextendr!r,rranyrr r&integer)r%r&rS_kvZsum_okr(rrrr ,s0    zIntegerPartition.__new__cCstt}|||j}|dgkr4t|jdiS|ddkr||dd8<|ddkrjd|d<qd||dd<|d<nv||dd8<|d|d}|d}d|d<x@|r|d8}||dkr||||7<||||8}qWt|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_keysrMrS)r.dkeysleftnewrrrprev_lexgs(     zIntegerPartition.prev_lexcCstt}|||j}|d}||jkrD||j|d<n@|dkr||dkr~||dd7<||d8<n8|d}||dd7<||d||d<d||<n||dkr0t|dkr|d||d<|j|d|d<n4|d}||d7<|||||d<d||<nT|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 rWrCrXrYr) rrZr[r\r]rSclearr#rM)r.r^rabZa1Zb1Zneedrrrnext_lexs>      zIntegerPartition.next_lexcCs8|jdkr2t|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)rgrrrrsz,IntegerPartition.as_dict..)_dictrr&r]rO)r.groupsrrrr\s  zIntegerPartition.as_dictcCsfd}t|jdg}|d}dg|}x:|dkr`x$|||krT|||d<|d8}q2W|d7}q(W|S)a Computes the conjugate partition of itself. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([6, 3, 3, 2, 1]) >>> a.conjugate [5, 4, 3, 1, 1, 1] rCr)rr&)r.rBZtemp_arrrUrerrr conjugates      zIntegerPartition.conjugatecCstt|jtt|jkS)aReturn True if self is less than other when the partition is listed from smallest to biggest. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([3, 1]) >>> a < a False >>> b = a.next_lex() >>> a < b True >>> a == b False )rreversedr&)r.r5rrrr;szIntegerPartition.__lt__cCstt|jtt|jkS)a Return True if self is less than other when the partition is listed from smallest to biggest. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> a = IntegerPartition([4]) >>> a <= a True )rrkr&)r.r5rrrr:s zIntegerPartition.__le__#csdfdd|jDS)a Prints the ferrer diagram of a partition. Examples ======== >>> from sympy.combinatorics.partitions import IntegerPartition >>> print(IntegerPartition([1, 1, 5]).as_ferrers()) ##### # #  csg|] }|qSrr)rr?)charrrrsz/IntegerPartition.as_ferrers..)joinr&)r.rnr)rnr as_ferrers s zIntegerPartition.as_ferrerscCstt|jS)N)strrr&)r.rrr__str__szIntegerPartition.__str__)N)rl)rGrHrIrJrhr]r rbrfr\rKrjr;r:rprrrrrrrMs ;%2  rMNcCsddlm}t|}|dkr$td||}g}x>|dkrn|d|}|d||}|||f|||8}q2W|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)_randintrCzn must be a positive integerT)rNcSsg|]\}}|g|qSrr)rrUmrrrr?sz,random_integer_partition..)Zsympy.utilities.randtestrsrrrEsortr)nZseedrsZrandintr&rUZmultrrrrandom_integer_partitions    rwcCst|d}x"td|dD]}d|d|f<qWxrtd|dD]`}xZt|D]N}|||kr|||d|f||d|df|||f<qNd|||f<qNWq@W|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 r )rtr^r?rBrrrRGS_generalizedCs  2rxcCs$|dkr dS|dkrdSt|SdS)a RGS_enum computes the total number of restricted growth strings possible for a superset of size m. Examples ======== >>> from sympy.combinatorics.partitions import RGS_enum >>> from sympy.combinatorics.partitions import Partition >>> RGS_enum(4) 15 >>> RGS_enum(5) 52 >>> RGS_enum(6) 203 We can check that the enumeration is correct by actually generating the partitions. Here, the 15 partitions of 4 items are generated: >>> a = Partition(list(range(4))) >>> s = set() >>> for i in range(20): ... s.add(a) ... a += 1 ... >>> assert len(s) == 15 rCrN)r )rtrrrr3cs r3cCs|dkrtd|dks$t||kr,tddg|d}d}t|}xptd|dD]^}||||f}||}||kr|d||<||8}|d7}qVt||d||<||;}qVWdd|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 argumentsrXcSsg|] }|dqS)rCr)rrrrrrszRGS_unrank..N)rr3rxr rZ)r-rtLrBDr?rVZcrrrrr2s"   r2cCslt|}d}t|}xRtd|D]D}t||dd}t|d|}||||df||7}q W|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#rxr rD)r@Zrgs_sizer-rzr?rvrtrrrr=s  r=)N)Z __future__rrZ sympy.corerrrZsympy.core.compatibilityrrr Z%sympy.functions.combinatorial.numbersr Zsympy.matricesr Zsympy.sets.setsr Zsympy.utilities.iterablesr rr collectionsrrrMrwrxr3r2r=rrrrs$     ' %#