A326373 Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) on n vertices.

1, 1, 1, 3, 435, 989555, 887050136795, 291072121058024908202443, 14704019422368226413236661148207899662350666147, 12553242487939461785560846872353486129110194529637343578112251094358919036718815137721635299

Offset

0,4

Comments

A set system (set of sets) is intersecting if no two edges are disjoint.

Links

Table of n, a(n) for n=0..9.

Formula

a(n) = A051185(n) - 1 - Sum_{k=1..n-1} binomial(n,k)*A000371(k). - Andrew Howroyd, Aug 12 2019

Example

The a(3) = 3 intersecting set systems with empty intersection:
  {}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

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

Crossrefs

The inverse binomial transform is the covering case A326364.

Set systems with empty intersection are A318129.

Intersecting set systems are A051185.

Intersecting antichains with empty intersection are A326366.

Cf. A000371, A006126, A007363, A014466, A058891, A305844, A307249, A318128, A326361, A326362, A326363, A326365.

Sequence in context: A269553 A361886 A086207 * A092052 A139999 A140870

Adjacent sequences: A326370 A326371 A326372 * A326374 A326375 A326376

Keyword

nonn

Author

Gus Wiseman, Jul 01 2019

Extensions

a(6)-a(9) from Andrew Howroyd, Aug 12 2019

Status

approved