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A326900 Number of set-systems on n vertices that are closed under union and intersection. 4
1, 2, 6, 29, 232, 3032, 62837, 2009408, 97034882, 6952703663, 728107141058, 109978369078580, 23682049666957359, 7195441649260733390, 3056891748255795885338, 1801430622263459795017565, 1462231768717868324127642932, 1624751185398704445629757084188, 2457871026957756859612862822442301 (list; graph; refs; listen; history; text; internal format)
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: A321961 A088957 A030538 * A181812 A330648 A074168

Adjacent sequences:  A326897 A326898 A326899 * A326901 A326902 A326903

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 04 2019

EXTENSIONS

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

STATUS

approved

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Last modified March 7 01:27 EST 2021. Contains 341859 sequences. (Running on oeis4.)