A326365 Number of intersecting antichains with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

1, 0, 0, 1, 23, 1834, 1367903, 229745722873, 423295077919493525420

Offset

0,5

Comments

Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

Links

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

Example

The a(4) = 23 intersecting antichains with empty intersection:
  {{1,2},{1,3},{2,3,4}}
  {{1,2},{1,4},{2,3,4}}
  {{1,2},{2,3},{1,3,4}}
  {{1,2},{2,4},{1,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,3},{1,2,4}}
  {{1,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{1,2,3}}
  {{1,4},{3,4},{1,2,3}}
  {{2,3},{2,4},{1,3,4}}
  {{2,3},{3,4},{1,2,4}}
  {{2,4},{3,4},{1,2,3}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Mathematica

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}], Or[Intersection[#1, #2]=={}, SubsetQ[#1, #2]]&], And[Union@@#==Range[n], #=={}||Intersection@@#=={}]&]], {n, 0, 4}]

Crossrefs

Intersecting antichain covers are A305844.

Intersecting covers with empty intersection are A326364.

Antichain covers with empty intersection are A305001.

The binomial transform is the non-covering case A326366.

Covering, intersecting antichains with empty intersection are A326365.

Cf. A006126, A007363, A014466, A051185, A058891, A305843, A307249, A318128, A318129, A326361, A326362, A326363.

Sequence in context: A183480 A002439 A229814 * A319508 A331340 A269122

Adjacent sequences: A326362 A326363 A326364 * A326366 A326367 A326368

Keyword

nonn,more

Author

Gus Wiseman, Jul 01 2019

Extensions

a(7)-a(8) from Andrew Howroyd, Aug 14 2019

Status

approved