A327354 Number of disconnected or empty antichains of nonempty subsets of {1..n} (non-spanning edge-connectivity 0).

1, 1, 2, 8, 53, 747, 45156, 54804920, 19317457655317

Offset

0,3

Comments

An antichain is a set of sets, none of which is a subset of any other.

The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

Links

Table of n, a(n) for n=0..8.

Formula

Equals the binomial transform of the exponential transform of A048143 minus A048143.

Example

The a(1) = 1 through a(3) = 8 antichains:
  {}  {}         {}
      {{1},{2}}  {{1},{2}}
                 {{1},{3}}
                 {{2},{3}}
                 {{1},{2,3}}
                 {{2},{1,3}}
                 {{3},{1,2}}
                 {{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], {1, n}], SubsetQ], Length[csm[#]]!=1&]], {n, 0, 4}]

Crossrefs

Column k = 0 of A327353.

The covering case is A120338.

The unlabeled version is A327426.

The spanning edge-connectivity version is A327352.

Cf. A014466, A326787, A327071, A327148, A327236, A327355, A327357.

Sequence in context: A195979 A203109 A197795 * A197931 A296788 A079503

Adjacent sequences: A327351 A327352 A327353 * A327355 A327356 A327357

Keyword

nonn,more

Author

Gus Wiseman, Sep 10 2019

Status

approved