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%I #21 Sep 30 2015 14:07:16
%S 0,1,2,5,31,163,1576
%N Number of conjugacy classes of two-element generating sets in the symmetric group S_n.
%C Two generating sets are considered to be the same if they differ only by some relabeling of the points, i.e., conjugating by some element of S_n. For instance, the generating set {(1,2), (1,2,3,4)} is the same as {(2,3),(1,2,3,4)} by the relabeling 1->2, 2->3, 3->4, 4->1. As a non-example, the generating sets {(1,2),(1,2,3,4,5)} and {(1,3),(1,2,3,4,5)} are different, since the points in the transpositions are differently placed in the 5-cycle.
%o (GAP)
%o # GAP 4.7 code for calculating the number of distinct 2-generating sets of
%o # symmetric groups.
%o # This code is written for readability, and to minimize package dependencies.
%o # 2015 _Attila Egri-Nagy_
%o # decides whether the given generating sets generate the symmetric group of
%o # degree n or not
%o IsSn := function(gens,n)
%o return Size(Group(gens))=Factorial(n);
%o end;
%o # returns all degree n permutations (i.e., elements of the symmetric group)
%o AllPermsDegn := function(n)
%o return AsList(SymmetricGroup(IsPermGroup,n));
%o end;
%o # first 5 entries of A001691 calculated in an inefficient manner
%o # taking all sets of cardinality 2 and check
%o gensets := List([1..5],
%o x->Filtered(Combinations(AllPermsDegn(x),2),
%o y->IsSn(y,x)));
%o Display(List(gensets,Size));
%o # returns the conjugacy class representative of P under G
%o # calculates the conjugacy class of P and returns the minimum element
%o # P - set of permutations
%o # G - permutation group
%o ConjClRep := function(P, G)
%o return Minimum(Set(AsList(G), x-> Set(P, y->y^x)));
%o end;
%o Display(List([1..5],
%o x->Size(Set(gensets[x],
%o y->ConjClRep(y,SymmetricGroup(IsPermGroup,x))))));
%Y Cf. A001691.
%K nonn,hard,more
%O 1,3
%A _Attila Egri-Nagy_, Aug 30 2015
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