\h+jSSKJr SSKJr \RrSrS SjrSrS SjrS Sjr S r S r g) ) Permutation)_distribute_gens_by_basecUVs1sHn[U5iM snUVs1sHn[U5iM sn:H$s snfs snf)a' Compare two lists of permutations as sets. Explanation =========== This is used for testing purposes. Since the array form of a permutation is currently a list, Permutation is not hashable and cannot be put into a set. Examples ======== >>> from sympy.combinatorics.permutations import Permutation >>> from sympy.combinatorics.testutil import _cmp_perm_lists >>> a = Permutation([0, 2, 3, 4, 1]) >>> b = Permutation([1, 2, 0, 4, 3]) >>> c = Permutation([3, 4, 0, 1, 2]) >>> ls1 = [a, b, c] >>> ls2 = [b, c, a] >>> _cmp_perm_lists(ls1, ls2) True )tuple)firstsecondas T/var/www/auris/envauris/lib/python3.13/site-packages/sympy/combinatorics/testutil.py_cmp_perm_listsr s@2$ $eE!He $$ %fE!Hf % && $ %s:?cL^ ^ SSKJn SSKJm [ US5(a[ UR SS95nURVs/sHoURPM snm U U 4Sjn/nU(d?UH7nU"U5(dMUR[R"U55 M9 U$UH#nU"U5(dMURU5 M% U$[ US5(a[X"U5U5$[ US 5(a[X"U/5U5$gs snf) NrPermutationGroup)_af_commutes_with generatorsTafc2>^[UU4SjT55$)Nc36># UHnT"TU5v M g7fN).0genrxs r <_naive_list_centralizer....Cs*UPT+)_naive_list_centralizer..Css*UPT*U'Ugetitem array_form) sympy.combinatorics.perm_groupsr sympy.combinatorics.permutationsrhasattrlistgenerate_diminor _array_formappendr_af_new_naive_list_centralizer) selfotherrrelementsrcommutes_with_genscentralizer_listelementrrs @@r r+r+$s@2Cul##,,,56','7'78'7! '78U#%g..$++K,?,?,HI$ $%g..$++G4$   " "&t-=e-DbII  % %&t-=ug-FKK &9sD!c SSKJn [X5nUn[[ U55HDnU"XF5nUR 5UR 5:wa gUR X5nMF UR 5S:wagg)a Verify the correctness of a base and strong generating set. Explanation =========== This is a naive implementation using the definition of a base and a strong generating set relative to it. There are other procedures for verifying a base and strong generating set, but this one will serve for more robust testing. Examples ======== >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> from sympy.combinatorics.testutil import _verify_bsgs >>> A = AlternatingGroup(4) >>> A.schreier_sims() >>> _verify_bsgs(A, A.base, A.strong_gens) True See Also ======== sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims rr FT)r#rrrangelenorder stabilizer)groupbaserrstrong_gens_distrcurrent_stabilizeri candidates r _verify_bsgsr>Ts8A0< 3t9 $%6%9:  # # %): :/::47C  !Q& r NcUcURU5n[URSS95n[XSS9n[ X45$)a Verify the centralizer of a group/set/element inside another group. This is used for testing ``.centralizer()`` from ``sympy.combinatorics.perm_groups`` Examples ======== >>> from sympy.combinatorics.named_groups import (SymmetricGroup, ... AlternatingGroup) >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.permutations import Permutation >>> from sympy.combinatorics.testutil import _verify_centralizer >>> S = SymmetricGroup(5) >>> A = AlternatingGroup(5) >>> centr = PermutationGroup([Permutation([0, 1, 2, 3, 4])]) >>> _verify_centralizer(S, A, centr) True See Also ======== _naive_list_centralizer, sympy.combinatorics.perm_groups.PermutationGroup.centralizer, _cmp_perm_lists Tr) centralizerr&r'r+r )r8argcentr centr_listcentr_list_naives r _verify_centralizerrE}sI: }!!#&e++t+45J.udC : 88r c^SSKJn UcURU5n[5n[ US5(a UR nO([ US5(aUnO[ US5(aU/nUR 5HmURU4SjW55 M U"[U55nURU5$)Nrr r __getitem__r"c3,># UH oT- v M g7frr)rrels r r)_verify_normal_closure..s9js(js) r#rnormal_closuresetr%rr'updater& is_subgroup)r8rAclosurer conjugates subgr_gens naive_closurerIs @r _verify_normal_closurerSs@.&&s+JsL!!^^ m $ $ l # #U ##%9j99&$T*%56M   } --r c zSSKJn SSKJnJn SSKJn /n[[U55H"n X9uppURX//U -U 45 M$ U"U6upnU"XUS- 5n[U[5(a SnU/nU/nO [U5n/n[U5H#n URU"XX)US- 55 M% U"U5nU"UVs/sHn[U5PM sn5n[URSS95nUR n[#5nURSS9H8nU"UU5nUH&n[%U"UU55nUR'U5 M( M: [U5nUR)5 S U-nUHnUS S US S :XaUS US :wa gUnM! [US5$s snf) a Canonicalize tensor formed by tensors of the different types. Explanation =========== sym_i symmetry under exchange of two component tensors of type `i` None no symmetry 0 commuting 1 anticommuting Parameters ========== g : Permutation representing the tensor. dummies : List of dummy indices. msym : Symmetry of the metric. v : A list of (base_i, gens_i, n_i, sym_i) for tensors of type `i`. base_i, gens_i BSGS for tensors of this type n_i number of tensors of type `i` Returns ======= Returns 0 if the tensor is zero, else returns the array form of the permutation representing the canonical form of the tensor. Examples ======== >>> from sympy.combinatorics.testutil import canonicalize_naive >>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs >>> from sympy.combinatorics import Permutation >>> g = Permutation([1, 3, 2, 0, 4, 5]) >>> base2, gens2 = get_symmetric_group_sgs(2) >>> canonicalize_naive(g, [2, 3], 0, (base2, gens2, 2, 0)) [0, 2, 1, 3, 4, 5] rr ) gens_products dummy_sgs)_af_rmulr3Tr)rN)r#rsympy.combinatorics.tensor_canrUrVr$rWr4r5r) isinstanceintextendrr&generater"rLraddsort)gdummiessymvrrUrVrWv1r<base_igens_in_isym_isizesbasesgensdgens num_typesSrDdliststshdqr prevs r canonicalize_naiverysNAG9 B 3q6]%&T" 6B48U34'+D gDF +E#s )eH E 9  Ywz364!8<=A%8%Q+a.%89A t$ %E A B ZZ4Z  QNAhq!n%A FF1I! RAFFH 9D  Sb6T#2Y uR   !:#9sF8cSSKJn SSKJnJn [ UR 55nURSSS9 UVs/sHoUSPM nnU"U5nSnUHupU[U 5- nM UV s/sHn /PM n n Sn UHNupU HCn XhXl:dMXURU 5 XU RU S-5 U S- n ME MP /n U HnU RU5 M [U 5U:XdeXUS-/- n US-n[U 5[ [U55:Xde[U 5n S/[U S5S--nU Hn U[U 5==S- ss'M /n[[U55H0n XnU(dMU"U 5unnURUUUS45 M2 UR5 [ [U55nU"U US/UQ76nU$s snfs sn f) a Return a certificate for the graph Parameters ========== gr : adjacency list Explanation =========== The graph is assumed to be unoriented and without external lines. Associate to each vertex of the graph a symmetric tensor with number of indices equal to the degree of the vertex; indices are contracted when they correspond to the same line of the graph. The canonical form of the tensor gives a certificate for the graph. This is not an efficient algorithm to get the certificate of a graph. Examples ======== >>> from sympy.combinatorics.testutil import graph_certificate >>> gr1 = {0:[1, 2, 3, 5], 1:[0, 2, 4], 2:[0, 1, 3, 4], 3:[0, 2, 4], 4:[1, 2, 3, 5], 5:[0, 4]} >>> gr2 = {0:[1, 5], 1:[0, 2, 3, 4], 2:[1, 3, 5], 3:[1, 2, 4, 5], 4:[1, 3, 5], 5:[0, 2, 3, 4]} >>> c1 = graph_certificate(gr1) >>> c2 = graph_certificate(gr2) >>> c1 [0, 2, 4, 6, 1, 8, 10, 12, 3, 14, 16, 18, 5, 9, 15, 7, 11, 17, 13, 19, 20, 21] >>> c1 == c2 True r) _af_invert)get_symmetric_group_sgs canonicalizec[US5$)Nr3)r5)rs r r#graph_certificate..=s S1Yr T)keyreverser3rX)r$r{r[r|r}r&itemsrar5r)r^sortedr4rr)grr{r|r}rrpvert num_indicesreneighr<verticesv2rbrkvlennr9rrccans r graph_certificaters F<T  E JJ&J5 !5aqT5E ! u EKs5z!  ""EqEH" ABx%)#q"))!,r#**1Q3/Q  A    q6[  {Q ''A ?D !9U4[) )) )AA 3HQK " #D SZA A 3t9  G 103JD$ HHdD!Q' (  IIK5%&G q'1 )q )C JO "#s G04 G5)Fr) sympy.combinatoricsrsympy.combinatorics.utilrrmulr r+r>rErSryrrr r rsA+=&:-L`&R!9H%.PK\Nr