A326900 Number of set-systems on n vertices that are closed under union and intersection.

1, 2, 6, 29, 232, 3032, 62837, 2009408, 97034882, 6952703663, 728107141058, 109978369078580, 23682049666957359, 7195441649260733390, 3056891748255795885338, 1801430622263459795017565, 1462231768717868324127642932, 1624751185398704445629757084188, 2457871026957756859612862822442301

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..18.

Example

The a(0) = 1 through a(3) = 29 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{2}}        {{2}}
             {{1,2}}      {{3}}
             {{1},{1,2}}  {{1,2}}
             {{2},{1,2}}  {{1,3}}
                          {{2,3}}
                          {{1,2,3}}
                          {{1},{1,2}}
                          {{1},{1,3}}
                          {{2},{1,2}}
                          {{2},{2,3}}
                          {{3},{1,3}}
                          {{3},{2,3}}
                          {{1},{1,2,3}}
                          {{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}]], SubsetQ[#, Union[Union@@@Tuples[#, 2], Intersection@@@Tuples[#, 2]]]&]], {n, 0, 3}]
(* Second program: *)
A006058 = Cases[Import["https://oeis.org/A006058/b006058.txt", "Table"], {_, _}][[All, 2]];
a[n_] := Sum[Binomial[n, k] A006058[[k + 1]], {k, 0, n}];
a /@ Range[0, 18] (* Jean-François Alcover, Jan 01 2020 *)

Crossrefs

Binomial transform of A006058 (the covering case).

The case closed under union only is A102896.

The case with {} allowed is A306445.

The BII-numbers of these set-systems are A326876.

The case closed under intersection only is A326901.

The unlabeled version is A326908.

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

Sequence in context: A088957 A030538 A348879 * A181812 A330648 A354418

Adjacent sequences: A326897 A326898 A326899 * A326901 A326902 A326903

Keyword

nonn

Author

Gus Wiseman, Aug 04 2019

Extensions

a(16)-a(18) from A006058 by Jean-François Alcover, Jan 01 2020

Status

approved