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

0, 1, 2, 5, 31, 163, 1576

Offset: 1

Author

Attila Egri-Nagy, Aug 30 2015

Keywords

• nonn
• hard
• more

Comments

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.

Crossrefs

Cf. A001691.

Programs

GAP

# GAP 4.7 code for calculating the number of distinct 2-generating sets of
# symmetric groups.
# This code is written for readability, and to minimize package dependencies.
# 2015 Attila Egri-Nagy
# decides whether the given generating sets generate the symmetric group of
# degree n or not
IsSn := function(gens,n)
  return Size(Group(gens))=Factorial(n);
end;
# returns all degree n permutations (i.e., elements of the symmetric group)
AllPermsDegn := function(n)
  return AsList(SymmetricGroup(IsPermGroup,n));
end;
# first 5 entries of A001691 calculated in an inefficient manner
# taking all sets of cardinality 2 and check
gensets := List([1..5],
                x->Filtered(Combinations(AllPermsDegn(x),2),
                        y->IsSn(y,x)));
Display(List(gensets,Size));
# returns the conjugacy class representative of P under G
# calculates the conjugacy class of P and returns the minimum element
# P - set of permutations
# G - permutation group
ConjClRep := function(P, G)
  return Minimum(Set(AsList(G), x-> Set(P, y->y^x)));
end;
Display(List([1..5],
        x->Size(Set(gensets[x],
                y->ConjClRep(y,SymmetricGroup(IsPermGroup,x))))));