A326903 Number of set-systems (without {}) on n vertices that are closed under intersection and have an edge containing all of the vertices, or Moore families without {}.

0, 1, 3, 16, 209, 11851, 8277238, 531787248525, 112701183758471199051

Offset

0,3

Comments

A set-system is a finite set of finite nonempty sets, so no two edges of such a set-system can be disjoint.

If {} is allowed, we get Moore families (A102896, cf A102895).

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) = A326901(n) / 2 for n > 0. - Andrew Howroyd, Aug 10 2019

Example

The a(1) = 1 through a(3) = 16 set-systems:
  {{1}}  {{1,2}}      {{1,2,3}}
         {{1},{1,2}}  {{1},{1,2,3}}
         {{2},{1,2}}  {{2},{1,2,3}}
                      {{3},{1,2,3}}
                      {{1,2},{1,2,3}}
                      {{1,3},{1,2,3}}
                      {{2,3},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1},{1,3},{1,2,3}}
                      {{2},{1,2},{1,2,3}}
                      {{2},{2,3},{1,2,3}}
                      {{3},{1,3},{1,2,3}}
                      {{3},{2,3},{1,2,3}}
                      {{1},{1,2},{1,3},{1,2,3}}
                      {{2},{1,2},{2,3},{1,2,3}}
                      {{3},{1,3},{2,3},{1,2,3}}

Mathematica

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

Crossrefs

The case closed under union and intersection is A006058.

The case with union instead of intersection is A102894.

The unlabeled version is A193674.

The case without requiring the maximum edge is A326901.

The covering case is A326902.

Cf. A000798, A001930, A102895, A102898, A326866, A326876, A326878, A326882, A326904.

Sequence in context: A196562 A317073 A272658 * A113597 A361366 A000273

Adjacent sequences: A326900 A326901 A326902 * A326904 A326905 A326906

Keyword

nonn,more

Author

Gus Wiseman, Aug 04 2019

Extensions

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

Status

approved