A327356 Number of connected separable antichains of nonempty sets covering n vertices (vertex-connectivity 1).

0, 0, 1, 3, 40, 1365

Offset

0,4

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.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

Links

Table of n, a(n) for n=0..5.

Example

Non-isomorphic representatives of the a(4) = 40 set-systems:
  {{1,2},{1,3,4}}
  {{1,2},{1,3},{1,4}}
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,3},{1,4},{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]]]];
vertConnSys[vts_, eds_]:=Min@@Length/@Select[Subsets[vts], Function[del, Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds, Alternatives@@del, {2}], {}]]!={Complement[vts, del]}]];
Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], SubsetQ], vertConnSys[Range[n], #]==1&]], {n, 0, 4}]

Crossrefs

Column k = 1 of A327351.

The graphical case is A327336.

The unlabeled version is A327436.

Cf. A014466, A048143, A307249, A326786, A326950, A327062, A327112, A327114, A327334.

Sequence in context: A012250 A094330 A110468 * A295612 A350545 A286661

Adjacent sequences: A327353 A327354 A327355 * A327357 A327358 A327359

Keyword

nonn,more

Author

Gus Wiseman, Sep 11 2019

Status

approved