A326901 Number of set-systems (without {}) on n vertices that are closed under intersection.

1, 2, 6, 32, 418, 23702, 16554476, 1063574497050, 225402367516942398102

Offset

0,2

Comments

A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.

Links

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

M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.

Formula

a(n) = 1 + Sum_{k=0, n-1} binomial(n,k)*A102895(k). - Andrew Howroyd, Aug 10 2019

Example

The a(3) = 32 set-systems:
  {}  {{1}}    {{1}{12}}    {{1}{12}{13}}   {{1}{12}{13}{123}}
      {{2}}    {{1}{13}}    {{2}{12}{23}}   {{2}{12}{23}{123}}
      {{3}}    {{2}{12}}    {{3}{13}{23}}   {{3}{13}{23}{123}}
      {{12}}   {{2}{23}}    {{1}{12}{123}}
      {{13}}   {{3}{13}}    {{1}{13}{123}}
      {{23}}   {{3}{23}}    {{2}{12}{123}}
      {{123}}  {{1}{123}}   {{2}{23}{123}}
               {{2}{123}}   {{3}{13}{123}}
               {{3}{123}}   {{3}{23}{123}}
               {{12}{123}}
               {{13}{123}}
               {{23}{123}}

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}]

Crossrefs

The case with union instead of intersection is A102896.

The case closed under union and intersection is A326900.

The covering case is A326902.

The connected case is A326903.

The unlabeled version is A326904.

The BII-numbers of these set-systems are A326905.

Cf. A000798, A001930, A006058, A102895, A102898, A182507, A326866, A326876, A326878, A326882.

Sequence in context: A232469 A034997 A067735 * A332537 A118077 A013976

Adjacent sequences: A326898 A326899 A326900 * A326902 A326903 A326904

Keyword

nonn,more

Author

Gus Wiseman, Aug 04 2019

Extensions

a(5)-a(8) from Andrew Howroyd, Aug 10 2019

Status

approved