+ i+l^RIHt^RIHt]PtRtR RltRtR RltR Rlt R t R t R#) ) Permutation)_distribute_gens_by_basecVUu0uFp\V4kK upVUu0uFp\V4kK up8H#uupiuupi)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&& |/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/combinatorics/testutil.py_cmp_perm_listsr s@2$ $eE!He $$ %fE!Hf % && $ %s8=c^a a ^RIHp^RIHo \ VR4'd\ VP RR74pVPUu.uFqUPNK upo V V 3Rlp.pV'gAVF8pV!V4'gKVP\P!V44K: V#VF$pV!V4'gKVPV4K& V#\ VR4'd\W!V4V4#\ VR4'd\W!V.4V4#R #uupi) rPermutationGroup)_af_commutes_with generatorsTafc~<a\;QJd VV3RlS4F 'dK R# R#!VV3RlS44#)c36<"TFpS!SV4xK R#5iN).0genrxs& r <_naive_list_centralizer....Cs*UPT+)_naive_list_centralizer..Cs*ss*UPT*Uss'Us'Us*UPT*U'Ugetitem array_formN) sympy.combinatorics.perm_groupsr sympy.combinatorics.permutationsrhasattrlistgenerate_diminor _array_formappendr_af_new_naive_list_centralizer) selfotherrrelementsrcommutes_with_genscentralizer_listelementrrs &&& @@r r+r+$s@2Cul##,,,56','7'78'7! '78U#%g..$++K,?,?,HI$ $%g..$++G4$   " "&t-=e-DbII  % %&t-=ug-FKK &9s D*c&^RIHp\W4pTp\\ V44FNpV!WF,4pVP 4VP 48wdR#VP W,4pKP VP 4^8wdR#R#)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 r FT)r#rrrangelenorder stabilizer)groupbaserrstrong_gens_distrcurrent_stabilizeri candidates&&& r _verify_bsgsr=Ts8A0< 3t9 $%6%9:  # # %): :/::47C  !Q& r NcVfVPV4p\VPRR74p\WRR7p\ W44#)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 )r7argcentr centr_listcentr_list_naives&&& r _verify_centralizerrD}sI: }!!#&e++t+45J.udC : 88r ca^RIHpVfVPV4p\4p\ VR4'dVP pM*\ VR4'dTpM\ VR4'dV.pVP 4FoVPV3RlX44K V!\V44pVPV4#)rr r __getitem__r"c34<"TF qS, xK R#5irr)rrels& r r)_verify_normal_closure..s9js((js) r#rnormal_closuresetr%rr'updater& is_subgroup)r7r@closurer conjugates subgr_gens naive_closurerHs&&& @r _verify_normal_closurerRs@.&&s+JsL!!^^ m $ $ l # #U ##%9j99&$T*%56M   } --r c ^RIHp^RIHpHp^RIHp.p\\V44F*p W9,wrrVPW..V ,V 34K, V!V!wrpV!WV^, 4p\V\4'd ^pV.pV.pM \V4p.p\V4F/p VPV!W,W),V^, 44K1 V!V4pT!VUu.uFp\V4NK up4p\VPRR74pVP p\#4pVPRR7F8pV!VV4pVF&p\%V!VV44pVP'V4K( K: \V4pVP)4RV,pVF*pVRRVRR8XdVR ,VR ,8wd^#TpK, \V^,4#uupi) a Canonicalize tensor formed by tensors of the different types. Explanation =========== sym_i symmetry under exchange of two component tensors of type `i` None no symmetry 0 commuting 1 anticommuting Parameters ========== g : Permutation representing the tensor. dummies : List of dummy indices. msym : Symmetry of the metric. v : A list of (base_i, gens_i, n_i, sym_i) for tensors of type `i`. base_i, gens_i BSGS for tensors of this type n_i number of tensors of type `i` Returns ======= Returns 0 if the tensor is zero, else returns the array form of the permutation representing the canonical form of the tensor. Examples ======== >>> from sympy.combinatorics.testutil import canonicalize_naive >>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs >>> from sympy.combinatorics import Permutation >>> g = Permutation([1, 3, 2, 0, 4, 5]) >>> base2, gens2 = get_symmetric_group_sgs(2) >>> canonicalize_naive(g, [2, 3], 0, (base2, gens2, 2, 0)) [0, 2, 1, 3, 4, 5] r ) gens_products dummy_sgs)_af_rmulTrN)r)r#rsympy.combinatorics.tensor_canrTrUr$rVr3r4r) isinstanceintextendrr&generater"rKraddsort)gdummiessymvrrTrUrVv1r;base_igens_in_isym_isizesbasesgensdgens num_typesSrDdliststshdqr prevs&&&* r canonicalize_naiverwsNAG9 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   !:#9s*G*c^RIHp^RIHpHp\ VP 44pVPRRR7VUu.uF qU^,NK ppV!V4p^pVFwrV\V 4, pK VU u.uFp .NK p p ^p VFowrV Fdp Wh,Wl,8gKWV,,PV 4WV ,,PV ^,4V ^, p Kf Kq .p V FpV PV4K \V 4V8XgQhWV^,., p V^,p\V 4\ \V448XgQh\V 4p ^.\V ^,4^,,pV F!p V\V 4;;,^, uu&K# .p\\V44F5p W,pV'gKV!V 4wppVPVVV^34K7 VP4\ \V44pV!V V^.VO5!pV#uupiuup i)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 ) _af_invert)get_symmetric_group_sgs canonicalizec&\V^,4#))r4)rs&r r#graph_certificate..=s S1Yr T)keyreverse)r$ryrYrzr{r&itemsr_r4r)r\sortedr3rr)grryrzr{rrpvert num_indicesrcneighr;verticesv2r`rivlennr8rracans& r graph_certificaters F<T  E JJ&J5 !5aqTT5E ! u EKs5z!  ""EqEH" ABx%)#q"))!,r#**1Q3/Q  A    q6[  {Q ''A ?D !9U4[) )) )AA 3HQK " #D SZA A 3t9  G 103JD$ HHdD!Q' (  IIK5%&G q'1 )q )C JO "#s I> I)Fr) sympy.combinatoricsrsympy.combinatorics.utilrrmulr r+r=rDrRrwrrr r rsA+=&:-L`&R!9H%.PK\Nr