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

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

Offset

1,3

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

Links

Table of n, a(n) for n=1..8.

Example

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

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 *)

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.

Sequence in context: A168129 A193500 A163885 * A064050 A162128 A337969

Adjacent sequences: A120335 A120336 A120337 * A120339 A120340 A120341

Keyword

nonn

Author

Greg Huber, Jun 22 2006

Status

approved