A263803 Number of conjugacy classes of independent sets of permutations of n points, i.e., subsets of the symmetric group of degree n up to relabeling the points with the property that none of the elements in the subset can be generated by the rest of the subset.
2, 3, 6, 31, 258, 10294
Offset: 1
Crossrefs
Cf. A263802.
Programs
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GAP
# GAP 4.7 http://www.gap-system.org # brute-force enumeration of conjugacy classes of # independent sets in the symmetric group, # inefficient (~4GB RAM needed, n=4 can take hours), # but short, readable, self-contained # higher terms can be calculated by the SubSemi package # https://github.com/egri-nagy/subsemi IsIndependentSet := function(A) return IsDuplicateFreeList(A) and (Size(A)<2 or ForAll(A,x-> not (x in Group(Difference(A,[x]))))); end; # we choose the minimal element (in lexicographic order) as the # representative of the equivalence class Rep := function(A, Sn) return Minimum(Set(Sn, g->Set(A, x->x^g))); end; CalcIndependentConjugacyClasses := function(n) local Sn, allsubsets, iss, reps; Sn := SymmetricGroup(IsPermGroup,n); allsubsets := Combinations(AsList(Sn)); iss := Filtered(allsubsets, IsIndependentSet); reps := Set(iss, x->Rep(x,Sn)); Print(Size(iss)," ", Size(reps),"\n"); end; for i in [1..4] do CalcIndependentConjugacyClasses(i); od;