D      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklm n o p q r s t u v w x y z { | } ~       5Type for permutations, considered as group elements. Wx .^ g returns the image of a vertex or point x under the action of the permutation g. 8 The dot is meant to be a mnemonic for point or vertex. Ub -^ g returns the image of an edge or block b under the action of the permutation g ? The dash is meant to be a mnemonic for edge or line or block. /Construct a permutation from a list of cycles.  For example, p [[1,2,3],[4,5]]K returns the permutation that sends 1 to 2, 2 to 3, 3 to 1, 4 to 5, 5 to 4 'A trick: g^-1 returns the inverse of g (g ~^ h returns the conjugate of g by h. Q The tilde is meant to a mnemonic, because conjugacy is an equivalence relation. Ox .^^ gs returns the orbit of the point or vertex x under the action of the gs Mb -^^ gs returns the orbit of the block or edge b under the action of the gs <_C n returns generators for Cn, the cyclic group of order n <_S n returns generators for Sn, the symmetric group on [1..n] >_A n returns generators for An, the alternating group on [1..n] SGiven generators for a group, return a (sorted) list of all elements of the group. U Implemented using a naive closure algorithm, so only suitable for small groups (|G| < 10000) VGiven generators for a group, return the order of the group (the number of elements). U Implemented using a naive closure algorithm, so only suitable for small groups (|G| < 10000) dconjClassReps gs returns a conjugacy class representatives and sizes for the group generated by gs. E This implementation is only suitable for use with small groups (|G| < 10000). M  !"#$ %& '()*+,-./0123456789:;<=>?@ABCDEFGHIJK  vGiven generators for a group, determine whether a permutation is a member of the group, using Schreier-Sims algorithm qGiven generators for a group, return a (sorted) list of all elements of the group, using Schreier-Sims algorithm tGiven generators for a group, return the order of the group (the number of elements), using Schreier-Sims algorithm LMNOPQRSTUVWXYZ[\]^_`ab LDatatype for graphs, represented as a list of vertices and a list of edges. g Both the list of vertices and the list of edges, and also the 2-element lists representing the edges, < are required to be in ascending order, without duplicates. 9combinationsOf k xs returns the subsets of xs of size k. K If xs is in ascending order, then the returned list is in ascending order =Safe constructor for graph from lists of vertices and edges. K graph (vs,es) checks that vs and es are valid before returning the graph. &c n is the cyclic graph on n vertices (k n is the complete graph on n vertices HGiven a graph with vertices which are lists of small integers, eg [1,2,3], f return a graph with vertices which are the numbers obtained by interpreting these as digits, eg 123. b The caller is responsible for ensuring that this makes sense (eg that the small integers are all < 10) :Given a graph with vertices which are lists of 0s and 1s, e return a graph with vertices which are the numbers obtained by interpreting these as binary digits.  For example, [1,1,0] -> 6. JThe diameter of a graph is maximum distance between two distinct vertices IThe girth of a graph is the size of the smallest cycle that it contains. M Note: If the graph contains no cycles, we return -1, representing infinity. -kneser n k returns the kneser graph KG n,k - 2 whose vertices are the k-element subsets of [1..n]&, with edges joining disjoint subsets >cdefghijklmnopqrstuvwxyz{|}~   ^Given a graph g, graphAuts g returns a strong generating set for the automorphism group of g.     !"#$%&'()*+,-./0123456789:;<=>?@ !"#$%&'()*+,-./0123456789:;<=>?@ ?@=><;:9876543210/.-,+*)('&%$#"! !"#$%&'()*+,-./0123456789:;<=>>?@@A/Phantom type for an elimination term ordering. u In the ordering, xis come before yjs come before zks, but within the xis, or yjs, or zks, grevlex ordering is used B0Phantom type representing grevlex term ordering C-Phantom type representing glex term ordering D,Phantom type representing lex term ordering EFGHABCDEFGHEFDCBAGHABCDEFFGH$IJKLMNOPQRSTUVWXYZ[\]^2F7 is a type for the finite field with 7 elements _2F5 is a type for the finite field with 5 elements `2F3 is a type for the finite field with 3 elements a2F2 is a type for the finite field with 2 elements bcdefgRQ is just the rationals, but with a better show function than the Prelude version hif2 lists the elements of F2 jf3 lists the elements of F3 kf5 lists the elements of F5 lf7 lists the elements of F7 ?IJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijkl$ghefbcdai`j_k^l]\[ZYXWVUTSRQPONMLKJI$IJKLMNOPQRSTUVWXYZ[\]^_`abcdcdeffghhijkl m#Type for multivariate polynomials. f ord is a phantom type defining how terms are ordered, r is the type of the ring we are working over. q For example, a common choice will be MPoly Grevlex Q, meaning polynomials over Q with the grevlex term ordering no*Create a variable with the supplied name. f By convention, variable names should usually be a single letter followed by none, one or two digits. pqrstuvwxyz{|}~*Convert a polynomial to lex term ordering +Convert a polynomial to glex term ordering .Convert a polynomial to grevlex term ordering 1mnopqrstuvwxyz{|}~mnoqrstuvwxyz{p}~|mnnopqrstuvwxyz{|}~ mGiven a list of polynomials over a field, return a Groebner basis for the ideal generated by the polynomials  F      !"#$  %&'()*+,-./01234567  89:;<=>?@ "ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrs tuvwxyz{|}~u  +" v returns the sum u+v of vectors u  -) v returns the difference u-v of vectors @k *> v returns the product k*v of the scalar k and the vector v u  .L v returns the dot product of vectors (also called inner or scalar product) u  *O v returns the tensor product of vectors (also called outer or matrix product) a  +# b returns the sum a+b of matrices a  -* b returns the difference a-b of matrices a  *' b returns the product a*b of matrices @k *> m returns the product k*m of the scalar k and the matrix m m  <*9 v is multiplication of a vector by a matrix on the left v  *;> m is multiplication of a vector by a matrix on the right !iMx n is the n*n identity matrix "jMx n is the n*n matrix of all 1s "zMx n is the n*n matrix of all 0s 5The inverse of a matrix (over a field), if it exists +The determinant of a matrix (over a field)  _ptsAG n fq returns the points of the affine geometry AG(n,Fq), where fq are the elements of Fq cptsPG n fq returns the points of the projective geometry PG(n,Fq), where fq are the elements of Fq PflatsPG n fq k returns the k-flats in PG(n,Fq), where fq are the elements of Fq PflatsAG n fq k returns the k-flats in AG(n,Fq), where fq are the elements of Fq  The lines (1-flats) in PG(n,fq)  The lines (1-flats) in AG(n,fq) [Incidence graph of PG(n,fq), considered as an incidence structure between points and lines [Incidence graph of AG(n,fq), considered as an incidence structure between points and lines  5'     " !"#$%&'()*+,-./0123 %456789:;<=>?@ABCDEFGHIJKLMNO PQRSTUVWXYZ[\]^_%`abcdefghijklmnopqrstuvwxyz{|}~ !"#$%&'()*+,-*+./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWWXYZ[\]^^_`abcdefghijklmnopqrstuvwxyz{|}}~~   3                                     ^ W     /*(      !"#$%&' ()*+5,6-.-/0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYABZ=[>?@\]^_`abcdefghijklmnopqrst+uvwxyz{|}~                                                                                              #      ! " # $ % & ' ( ) * + ,  - . /  0 1 2 3456789:;<=>?@ABC DEFGHIJKLMNOPQjiRSTUVWXKYZ[\]^_`abcdef#ghijklmnopqrstuvwxyz{|}~cpcd      Z !"#$%&'HaskellForMaths-0.1.8"Math.Algebra.Group.StringRewriting#Math.Algebra.Group.PermutationGroupMath.Algebra.Group.SchreierSimsMath.Combinatorics.GraphMath.Combinatorics.GraphAutsMath.Common.IntegerAsType!Math.Algebra.Commutative.MonomialMath.Algebra.Field.BaseMath.Algebra.Commutative.MPolyMath.Algebra.Commutative.GBasisMath.Algebra.Field.Extension"Math.Algebra.NonCommutative.NCPoly)Math.Algebra.NonCommutative.TensorAlgebra%Math.Projects.KnotTheory.LaurentMPolyMath.Projects.KnotTheory.Braid&Math.Projects.KnotTheory.TemperleyLieb%Math.Projects.KnotTheory.IwahoriHeckeMath.Algebra.LinearAlgebra!Math.Combinatorics.FiniteGeometryMath.Combinatorics.Design'Math.Combinatorics.StronglyRegularGraphMath.Combinatorics.HypergraphMath.Projects.RootSystem(Math.Projects.ChevalleyGroup.ExceptionalMath.Common.ListSetMath.Projects.Rubik#Math.Algebra.NonCommutative.GSBasis&Math.Projects.ChevalleyGroup.ClassicalS PermutationP.^-^p^-~^.^^-^^_C_S_Aeltsorder conjClassRepsisMemberGraphGcombinationsOfgraph nullGraphck fromDigits fromBinarydiametergirthkneser graphAutsT97T89T83T79T73T71T67T61T59T53T47T43T41T37T31T29T23T19T17T13T11T7T5T3T2TOneTZeroTMinus1M IntegerAsTypevalueElimGrevlexGlexLexMonomialconvertMsupportMF97F89F83F79F73F71F67F61F59F53F47F43F41F37F31F29F23F19F17F13F11F7F5F3F2 FiniteFieldeltsFqbasisFqFpQf2f3f5f7MPolyMPvarabdstuvwxyzx0x1x2x3toLextoGlex toGrevlextoElimgb QSqrtMinus5 QSqrtMinus3 QSqrtMinus2 QSqrtMinus1QSqrt7QSqrt5QSqrt3QSqrt2SqrtF32 ConwayF32F27 ConwayF27F25 ConwayF25F16 ConwayF16F9ConwayF9F8ConwayF8F4ConwayF4ExtensionFieldExtPolynomialAsTypepvalueUPolyUP quotRemUP conjugate InvertibleinvVarZYXNPolyNPWeylGensDBasisE LaurentMPolyLPLaurentMonomialLM BraidGensLPQTemperleyLiebGensIwahoriHeckeGensT<+><->*><.><*><<+>><<->><<*>>*>><<*><*>>iMxjMxzMxinversereducedRowEchelonFormdet ZeroOneStarStarOneZeroptsAGptsPGflatsPGflatsAGlinesPGlinesAGincidenceGraphPGincidenceGraphAGDesign DesignVertexBC Incidence HypergraphHTypeFAbasisEltOctonionOrewrite rewrite''splitSubstring findOverlap knuthBendix1ordpairshortlex knuthBendix2merge knuthBendix3 knuthBendixnfss_s1s2s3tri_D toListSet isListSetunion intersect\\symDiffdisjointisSubsetrotateL fromPairs fromPairs'toPairsfromListsupp fromCyclestoCyclescycleOfparitysignorderEltcommclosureSclosureorbitorbitPorbitVorbitBorbitEactionorbits_D2dpwrtoSn fromDigits' fromBinary'eltsSminsupporderTGSeltsTGS tgsFromSgsorderSGSgens~^^ conjClass centralizercentre normalizer stabilizerptStabsetStab reduceGens normalClosure commutatorGp derivedSubgpisSubgpisNormal**^cosets cosetAction quotientGp//~~^conjugateSubgps subgpAction cosetRepsGxschreierGeneratorsGxsiftfindBasesgsbsgsbsgs'newLevel newLevel'ssss' isMemberBSGSeltsBSGScartProd orderBSGSindexreduceGensBSGSsetpowerset isSetSystemisGraphtoGraphverticesedgesincidenceMatrixfromIncidenceMatrixadjacencyMatrixfromAdjacencyMatrixkbkb'q'q tetrahedroncube octahedron dodecahedron icosahedronto1npetersen complement lineGraph lineGraph'sizevalency valencies regularParam isRegularisCubicnbrs findPathsdistance findCyclesdistancePartition component isConnectedjjohnsonbipartiteKneser desargues1gp petersen2prismdurer mobiusKantor dodecahedron2 desargues2isVertexTransitiveisEdgeTransitive->^isArcTransitiveisArcTransitive'findArcsisnArcTransitiveis2ArcTransitiveis3ArcTransitiveisDistanceTransitiverefine isGraphAut graphAuts1 graphAuts2 graphAuts3 isSingleton incidenceAuts removeGens graphAutsNew graphIsosisIsoflrdiffsdegMdividesMproperlyDividesMlcmMgcdMcoprimeM numeratorQ denominatorQextendedEuclid primitiveEltpowerscharf11f13f17f19f23f29f31f37f41f43f47f53f59f61f67f71f73f79f83f89f97cmpTerm mergeTermscollectx_ convertMPvarLexvarElimltlmdegmulTdivTdividesTproperlyDividesTlcmT.* quotRemMP%%divModMPdivMPmodMPinjecttoMonictoZsubstsupportsPolyisGBgb1pairWithreducegb2pairs!gb2bgb3bcmpFstmergeBygb4bsugar cmpNormalcmpSuggb3gb4showUPtoUPolyconvertmonomialmodUPextendedEuclidUP/>embedpolysf4a4f8a8f9a9f16a16f25a25f27a27f32a32 frobeniusAutdegreesqrt2sqrt3sqrt5sqrt7i sqrtminus2 sqrtminus3 sqrtminus5divMlc quotRemNPremNPremNP2gb'gb2'mbasisQAe_e1e2e3e4dim tensorBasis extRelationsextnf exteriorBasis symRelationssymnfsymmetricBasis weylRelationsweylnf weylBasisd_d1d2d3deltaweylRelations'weylnf' weylBasis'degLM denominatorLMlcmLMdivLM quotRemLPreduceLP denominatorLPsqrtvar^^^s4writhek3_1k4_1k5_1k7_1d' tlRelationsdimTLtlnftlBasistr'tra' fromBraidjonest_t1t2t3t4z' ihRelationsdimIHihnfihBasistau'tauhomflymhomfly'homfly''coeffsjones'fMatrixfMatrix'idMxinverse1inverse2rowEchelonFormkernel kernelRREpnfispnf closureAGlineAG closurePGqtorialqnomial numFlatsPG numFlatsAGqtorialsqnomialsrrefslinesAG1linesAG2orderGLorderAffdesigntoDesignisValidpointsblocksnoRepeatedBlocks tDesignParamsfindvk findlambda designParams isStructureisDesign is2DesignisSquare subsetDesign pairDesignag2pg2 flatsDesignPGpg flatsDesignAGag paleyDesign fanoPlanedual derivedDesign pointResidualcomplementaryDesign blockResidual isDesignAutincidenceGraph designAuts designAuts1 designAuts2 designAutsNew alphaL2_23 betaL2_23 gammaL2_23l2_23deltaM24m24m24sgsm23sgsm22sgsoctads_5_8_24s_4_7_23s_3_6_22 srgParamsisSRGt'l2l2' paleyGraphclebschclebsch'clebsch2triplesheptads+^+^^hoffmanSingletonhoffmanSingleton' inducedA7hsA7gewirtzgewirtz'higmanSimsGraphhigmanSimsGraph' inducedM22 higmanSimsM22_HS2_HSsp2spswitchschlafli schlafli' mcLaughlin mcLaughlin'_McL2_McL hypergraph toHypergraph isUniformsame fromGraph fromDesignisPartialLinearSpaceisProjectivePlaneisProjectivePlaneTriisProjectivePlaneQuadisGeneralizedQuadranglegriddualGridisGenQuadrangle'isConfiguration heawoodGraphdesarguesConfigurationdesarguesGraphpappusConfiguration pappusGraph coxeterGraphduads synthemestutteCoxeterGraphintersectionGraph simpleSystem weylPerms weylMatriceswMx cartanMatrixsetDiagdynkinFromCartan dynkinDiagramcoxeterFromDynkin coxeterMatrixfromCoxeterMatrixfromCoxeterMatrix2coxeterPresentation eltsCoxeter poincarePolyelemMxlieMult+|++-+form rootSystemnumRoots orderWeyl factorialtest1test2numPtsAGnumPtsPGslelemTransvectionorderLorderS2orderS omegaevenomegaoddoi0i1i2i3i4i5i6toListexposenfconjsqnorm isOrthogonal antiCommutes octonionsisUnitunitImagOctonionsautFrom%^alpha3beta3gamma3sgamma3alpha3'beta3'gamma3'alpha4beta4gamma4sgamma4alpha4'beta4'gamma4'