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A261750 Number of conjugacy classes of two-element generating sets in the symmetric group S_n. 0


%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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Last modified November 30 21:57 EST 2022. Contains 358453 sequences. (Running on oeis4.)