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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.
0
2, 3, 6, 31, 258, 10294
OFFSET
1,1
PROG
(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;
CROSSREFS
Cf. A263802.
Sequence in context: A082611 A033689 A018324 * A178175 A351883 A129379
KEYWORD
nonn,hard,more
AUTHOR
Attila Egri-Nagy, Oct 27 2015
STATUS
approved