A322063 Number of ways to choose a stable partition of an antichain of sets spanning n vertices.

1, 1, 3, 25, 773, 160105

Offset: 0

Author

Gus Wiseman, Nov 25 2018

Keywords

• nonn
• more

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.

Examples

			The a(3) = 25 stable partitions of antichains on 3 vertices. The antichain is on top, and below is a list of all its stable partitions.
  {1}{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},{2,3}}    {{1,2},{3}}    {{1,3},{2}}    {{1,2},{3}}    {{1,3},{2}}
  {{1,2},{3}}    {{1,3},{2}}    {{1},{2},{3}}  {{1},{2},{3}}  {{1},{2},{3}}
  {{1,3},{2}}    {{1},{2},{3}}
  {{1},{2},{3}}
.
  {1,3}{2,3}     {1,2}{2,3}     {1,2}{1,3}     {1,2}{1,3}{2,3}
  --------       --------       --------       --------
  {{1,2},{3}}    {{1,3},{2}}    {{1},{2,3}}    {{1},{2},{3}}
  {{1},{2},{3}}  {{1},{2},{3}}  {{1},{2},{3}}
		

Crossrefs

Cf. A000110, A000569, A006125, A006126, A229048, A240936, A277203, A321979, A322064, A322065.

Programs

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]]]];
Table[Sum[Length[stableSets[Complement[Subsets[Range[n]],Union@@Subsets/@stn],SubsetQ]],{stn,sps[Range[n]]}],{n,5}]