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A322065
Number of ways to choose a stable partition of a connected antichain of sets spanning n vertices.
2
1, 1, 1, 11, 525, 146513
OFFSET
0,4
COMMENTS
A stable partition of a hypergraph or set system is a set partition of the vertices where no non-singleton edge has all its vertices in the same block.
EXAMPLE
The a(3) = 11 stable partitions. The connected antichain is on top, and below is a list of all its stable partitions.
{1,2,3} {1,3}{2,3} {1,2}{2,3} {1,2}{1,3} {1,2}{1,3}{2,3}
-------- -------- -------- -------- --------
{{1},{2,3}} {{1,2},{3}} {{1,3},{2}} {{1},{2,3}} {{1},{2},{3}}
{{1,2},{3}} {{1},{2},{3}} {{1},{2},{3}} {{1},{2},{3}}
{{1,3},{2}}
{{1},{2},{3}}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
stableSets[u_, Q_]:=If[Length[u]===0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r===w||Q[r, w]||Q[w, r]], Q]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Sum[Length[Select[stableSets[Complement[Subsets[Range[n]], Union@@Subsets/@stn], SubsetQ], And[Union@@#==Range[n], Length[csm[#]]==1]&]], {stn, sps[Range[n]]}], {n, 5}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 25 2018
STATUS
approved