(GAP)
# GAP 4.7 code for calculating the number of distinct 2generating sets of
# symmetric groups.
# This code is written for readability, and to minimize package dependencies.
# 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))))));
