A120338 - OEIS

%I #10 Sep 27 2019 08:45:01

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

%N Number of disconnected antichain covers of a labeled n-set.

%C 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

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

%t 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]]]]]]]]];

%t 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]]]];

%t Table[Length[Select[stableSets[Subsets[Range[n]],SubsetQ],Union@@#==Range[n]&&Length[csm[#]]!=1&]],{n,4}] (* _Gus Wiseman_, Sep 26 2019 *)

%Y Cf. A006126, A007153, A048143, A327355.

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

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

%Y The non-covering version is A327354.

%Y The unlabeled version is A327426.

%K nonn

%O 1,3

%A _Greg Huber_, Jun 22 2006