A322065 Number of ways to choose a stable partition of a connected antichain of sets spanning n vertices.

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.

Links

Table of n, a(n) for n=0..5.

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}]

Crossrefs

Cf. A000110, A001187, A006125, A048143, A229048, A240936, A245883, A277203, A321911, A321979, A322063, A322064.

Sequence in context: A142809 A139198 A004800 * A065823 A233198 A358162

Adjacent sequences: A322062 A322063 A322064 * A322066 A322067 A322068

Keyword

nonn,more

Author

Gus Wiseman, Nov 25 2018

Status

approved