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

1, 1, 3, 25, 773, 160105

Offset

0,3

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) = 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}}

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}]

Crossrefs

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

Sequence in context: A062411 A136516 A002021 * A350902 A306792 A012764

Adjacent sequences: A322060 A322061 A322062 * A322064 A322065 A322066

Keyword

nonn,more

Author

Gus Wiseman, Nov 25 2018

Status

approved