cp's OEIS Frontend

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

%I A261750 #21 Sep 30 2015 14:07:16
%S A261750 0,1,2,5,31,163,1576
%N A261750 Number of conjugacy classes of two-element generating sets in the symmetric group S_n.
%C A261750 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 A261750 (GAP)
%o A261750 # GAP 4.7 code for calculating the number of distinct 2-generating sets of
%o A261750 # symmetric groups.
%o A261750 # This code is written for readability, and to minimize package dependencies.
%o A261750 # 2015 _Attila Egri-Nagy_
%o A261750 # decides whether the given generating sets generate the symmetric group of
%o A261750 # degree n or not
%o A261750 IsSn := function(gens,n)
%o A261750   return Size(Group(gens))=Factorial(n);
%o A261750 end;
%o A261750 # returns all degree n permutations (i.e., elements of the symmetric group)
%o A261750 AllPermsDegn := function(n)
%o A261750   return AsList(SymmetricGroup(IsPermGroup,n));
%o A261750 end;
%o A261750 # first 5 entries of A001691 calculated in an inefficient manner
%o A261750 # taking all sets of cardinality 2 and check
%o A261750 gensets := List([1..5],
%o A261750                 x->Filtered(Combinations(AllPermsDegn(x),2),
%o A261750                         y->IsSn(y,x)));
%o A261750 Display(List(gensets,Size));
%o A261750 # returns the conjugacy class representative of P under G
%o A261750 # calculates the conjugacy class of P and returns the minimum element
%o A261750 # P - set of permutations
%o A261750 # G - permutation group
%o A261750 ConjClRep := function(P, G)
%o A261750   return Minimum(Set(AsList(G), x-> Set(P, y->y^x)));
%o A261750 end;
%o A261750 Display(List([1..5],
%o A261750         x->Size(Set(gensets[x],
%o A261750                 y->ConjClRep(y,SymmetricGroup(IsPermGroup,x))))));
%Y A261750 Cf. A001691.
%K A261750 nonn,hard,more
%O A261750 1,3
%A A261750 _Attila Egri-Nagy_, Aug 30 2015