\h>6SSKJr SSKJr "SS5rSrg) combinations)GrayCodec8\rSrSrSrSrSrSrSrSr Sr Sr Sr Sr SrS rS rS rS rS r\S5r\S5r\S5r\S5r\S5r\S5r\S5r\S5r\S5r\S5r\S5r \S5r!\S5r"Sr#g)Subseta Represents a basic subset object. Explanation =========== We generate subsets using essentially two techniques, binary enumeration and lexicographic enumeration. The Subset class takes two arguments, the first one describes the initial subset to consider and the second describes the superset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.next_binary().subset ['b'] >>> a.prev_binary().subset ['c'] Nc[U5[U5:a [S5eUH"nX2;dM [SRU55e [R U5nXlX$lU$)a Default constructor. It takes the ``subset`` and its ``superset`` as its parameters. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.subset ['c', 'd'] >>> a.superset ['a', 'b', 'c', 'd'] >>> a.size 2 zRInvalid arguments have been provided. The superset must be larger than the subset.zFThe superset provided is invalid as it does not contain the element {})len ValueErrorformatobject__new___subset _superset)clssubsetsupersetelemobjs S/var/www/auris/envauris/lib/python3.13/site-packages/sympy/combinatorics/subsets.pyrSubset.__new__$su$ v;X &HI ID# ">>DfTlLLnnS!   c[U[5(d[$URUR:H=(a URUR:H$)zReturn a boolean indicating whether a == b on the basis of whether both objects are of the class Subset and if the values of the subset and superset attributes are the same. ) isinstancerNotImplementedrr)selfothers r__eq__ Subset.__eq__Bs< %((! !{{ell*Nt}}/NNrcB[RURUR5n[ SR U5S5U-SUR --n[U5SSRUR S5n[RURU5$)a This is a helper function. It iterates over the binary subsets by ``k`` steps. This variable can be both positive or negative. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.iterate_binary(-2).subset ['d'] >>> a = Subset(['a', 'b', 'c'], ['a', 'b', 'c', 'd']) >>> a.iterate_binary(2).subset [] See Also ======== next_binary, prev_binary N0) rbitlist_from_subsetrrintjoin superset_sizebinrjustsubset_from_bitlist)rkbin_listnbitss riterate_binarySubset.iterate_binaryKs,--dkk4==I "A & *a1C1C.C C1vabz 2 2C8))$-->>rc$URS5$)aE Generates the next binary ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.next_binary().subset ['b'] >>> a = Subset(['a', 'b', 'c', 'd'], ['a', 'b', 'c', 'd']) >>> a.next_binary().subset [] See Also ======== prev_binary, iterate_binary r/rs r next_binarySubset.next_binaryfs(""1%%rc$URS5$)aI Generates the previous binary ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([], ['a', 'b', 'c', 'd']) >>> a.prev_binary().subset ['a', 'b', 'c', 'd'] >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.prev_binary().subset ['c'] See Also ======== next_binary, iterate_binary r3r4s r prev_binarySubset.prev_binary|s(""2&&rcDURS- n[RURUR5nX;a}US- U;aUR US- 5 OUR U5 US- nUS:aX;aUS- nUS:aX;aMUS:a%UR U5 UR US-5 O1X;aUS:aUS- nX;aUS:aMUR US-5 /nURnUHnUR XA5 M [X45$)aF Generates the next lexicographically ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.next_lexicographic().subset ['d'] >>> a = Subset(['d'], ['a', 'b', 'c', 'd']) >>> a.next_lexicographic().subset [] See Also ======== prev_lexicographic r2rr'rsubset_indicesrrremoveappendriindicesret_set super_sets rnext_lexicographicSubset.next_lexicographics(    "'' T]]C <1uq1u%q!E1f!1AA1f!16NN1%NN1Q3'"qAvE"qAv NN1q5 !MM A NN9< (g))rcURS- n[RURUR5nUS:aX;aUS- nUS:aX;aMUS:Xd US- U;aUR U5 OIUS:a%UR U5 UR US- 5 UR URS- 5 /nURnUHnUR XA5 M [X45$)aI Generates the previous lexicographically ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([], ['a', 'b', 'c', 'd']) >>> a.prev_lexicographic().subset ['d'] >>> a = Subset(['c','d'], ['a', 'b', 'c', 'd']) >>> a.prev_lexicographic().subset ['c'] See Also ======== next_lexicographic r2rr<r@s rprev_lexicographicSubset.prev_lexicographics(    "'' T]]C1f)AA1f) 6QUg% NN1 Avq!q1u% NN4--1 2MM A NN9< (g))rc[R"URURU-UR-5n[ R URU5$)aV Helper function used for prev_gray and next_gray. It performs ``k`` step overs to get the respective Gray codes. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([1, 2, 3], [1, 2, 3, 4]) >>> a.iterate_graycode(3).subset [1, 4] >>> a.iterate_graycode(-2).subset [1, 2, 4] See Also ======== next_gray, prev_gray )runrankr' rank_gray cardinalityrr*r)rr+ unranked_codes riterate_graycodeSubset.iterate_graycodesN(!(:(:(,(:d>N>N'NP ))$--*79 9rc$URS5$)z Generates the next Gray code ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([1, 2, 3], [1, 2, 3, 4]) >>> a.next_gray().subset [1, 3] See Also ======== iterate_graycode, prev_gray r2rOr4s r next_graySubset.next_grays"$$Q''rc$URS5$)z Generates the previous Gray code ordered subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([2, 3, 4], [1, 2, 3, 4, 5]) >>> a.prev_gray().subset [2, 3, 4, 5] See Also ======== iterate_graycode, next_gray r8rRr4s r prev_graySubset.prev_grays"$$R((rcURcH[SR[R UR UR 55S5UlUR$)a Computes the binary ordered rank. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset([], ['a','b','c','d']) >>> a.rank_binary 0 >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.rank_binary 3 See Also ======== iterate_binary, unrank_binary r!r") _rank_binaryr%r&rr$rrr4s r rank_binarySubset.rank_binary&sT*    $ #BGG**4;;+/==:%;<=!?D    rc^URcIU4Sjm[RURUR5nT"XSUR 5UlUR$)a  Computes the lexicographic ranking of the subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.rank_lexicographic 14 >>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6]) >>> a.rank_lexicographic 43 c>U/:XdX#:agX!;a!URU5 ST"XUS-U5-$SX2- S- -T"XUS-U5-$)Nrr2r")r>)r subset_indexrAr-_ranklexs rr_+Subset.rank_lexicographic.._ranklexRs`2%$ ''*xAE1EEE1519~QUA(NNNrr) _rank_lexrr=rrr')rrBr_s @rrank_lexicographicSubset.rank_lexicographicAsQ >> ! O++DKKGG%dQ8J8JKDN~~rcURcL[RURUR5n[ [ U5US9RUlUR$)a  Computes the Gray code ranking of the subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c','d'], ['a','b','c','d']) >>> a.rank_gray 2 >>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6]) >>> a.rank_gray 27 See Also ======== iterate_graycode, unrank_gray )start)_rank_graycoderr$rrrr rank)rr.s rrLSubset.rank_gray]sO*    &--dkk4==ID"*3t9D"A"F"FD """rcUR$)z Gets the subset represented by the current instance. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.subset ['c', 'd'] See Also ======== superset, size, superset_size, cardinality )rr4s rr Subset.subsetws$||rc,[UR5$)z Gets the size of the subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.size 2 See Also ======== subset, superset, superset_size, cardinality )r rr4s rsize Subset.sizes$4;;rcUR$)z Gets the superset of the subset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.superset ['a', 'b', 'c', 'd'] See Also ======== subset, size, superset_size, cardinality )rr4s rrSubset.supersets$~~rc,[UR5$)z Returns the size of the superset. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.superset_size 4 See Also ======== subset, superset, size, cardinality )r rr4s rr'Subset.superset_sizes$4==!!rc SUR-$)z Returns the number of all possible subsets. Examples ======== >>> from sympy.combinatorics import Subset >>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd']) >>> a.cardinality 16 See Also ======== subset, superset, size, superset_size r")r'r4s rrMSubset.cardinalitys$4%%&&rc[U5[U5:wa [S5e/n[[U55H nX$S:XdM URX5 M" [ X15$)z Gets the subset defined by the bitlist. Examples ======== >>> from sympy.combinatorics import Subset >>> Subset.subset_from_bitlist(['a', 'b', 'c', 'd'], '0011').subset ['c', 'd'] See Also ======== bitlist_from_subset z$The sizes of the lists are not equal1)r r ranger?r)rrDbitlistrCrAs rr*Subset.subset_from_bitlists]" y>S\ )CD Ds7|$AzS y|,%g))rcS/[U5-n[U[5(a URn[R X5HnSX4'M SR U5$)z Gets the bitlist corresponding to a subset. Examples ======== >>> from sympy.combinatorics import Subset >>> Subset.bitlist_from_subset(['c', 'd'], ['a', 'b', 'c', 'd']) '0011' See Also ======== subset_from_bitlist r#rur!)r rrrr=r&)rrrrwrAs rr$Subset.bitlist_from_subsetsU"%#h-' ff % %]]F&&v8AGJ9wwwrcz[U5SSR[U5S5n[R X#5$)z Gets the binary ordered subset of the specified rank. Examples ======== >>> from sympy.combinatorics import Subset >>> Subset.unrank_binary(4, ['a', 'b', 'c', 'd']).subset ['b'] See Also ======== iterate_binary, rank_binary r"Nr#)r(r)r rr*)rrgrr.s r unrank_binarySubset.unrank_binary s5"4y}""3x=#6))(99rcl[R"[U5U5n[R X#5$)a Gets the Gray code ordered subset of the specified rank. Examples ======== >>> from sympy.combinatorics import Subset >>> Subset.unrank_gray(4, ['a', 'b', 'c']).subset ['a', 'b'] >>> Subset.unrank_gray(0, ['a', 'b', 'c']).subset [] See Also ======== iterate_graycode, rank_gray )rrKr rr*)rrgrgraycode_bitlists r unrank_graySubset.unrank_gray s*&$??3x=$?))(EErcX!pC[U5n0n[U5H*upxX;dM XvU'URU5 U(aM* O /$UV s/sHoU PM sn $s sn f)aReturn indices of subset in superset in a list; the list is empty if all elements of ``subset`` are not in ``superset``. Examples ======== >>> from sympy.combinatorics import Subset >>> superset = [1, 3, 2, 5, 4] >>> Subset.subset_indices([3, 2, 1], superset) [1, 2, 0] >>> Subset.subset_indices([1, 6], superset) [] >>> Subset.subset_indices([], superset) [] )set enumerater>) rrrabsbdrAaibis rr=Subset.subset_indices6sg$1 V q\EAx" " r "I !""""""sA )rYrfra)$__name__ __module__ __qualname____firstlineno____doc__rYrarfrrrrr/r5r9rErHrOrSrVpropertyrZrbrLrrlrr'rM classmethodr*r$r|rr=__static_attributes__rrrrs].LINGI<O?6&,',+*Z&*P92(&)&!!46##2&  &&""&''&**0  .::&FF*##rrc[X5$)aw Finds the subsets of size ``k`` in lexicographic order. This uses the itertools generator. Examples ======== >>> from sympy.combinatorics.subsets import ksubsets >>> list(ksubsets([1, 2, 3], 2)) [(1, 2), (1, 3), (2, 3)] >>> list(ksubsets([1, 2, 3, 4, 5], 2)) [(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)] See Also ======== Subset r)rr+s rksubsetsrVs*  $$rN) itertoolsrsympy.combinatorics.graycoderrrrrrrs"1M #M #`%r