A120338 Number of disconnected antichain covers of a labeled n-set.

0, 1, 4, 30, 546, 41334, 54502904, 19317020441804

Offset: 1

Author

Greg Huber, Jun 22 2006

Keywords

• nonn

Comments

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices. - Gus Wiseman, Sep 26 2019

Examples

			a(3)=4: the four disconnected covers are {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}} and {{1},{2},{3}}.
		

Crossrefs

Cf. A006126, A007153, A048143, A327355.

Column k = 0 of A327351, if we assume a(0) = 1.

Column k = 0 of A327357, if we assume a(0) = 1.

The non-covering version is A327354.

The unlabeled version is A327426.

Programs

Mathematica

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
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[Length[Select[stableSets[Subsets[Range[n]],SubsetQ],Union@@#==Range[n]&&Length[csm[#]]!=1&]],{n,4}] (* Gus Wiseman, Sep 26 2019 *)