A326900 - OEIS

%I #11 Jan 29 2024 13:48:34

%S 1,2,6,29,232,3032,62837,2009408,97034882,6952703663,728107141058,

%T 109978369078580,23682049666957359,7195441649260733390,

%U 3056891748255795885338,1801430622263459795017565,1462231768717868324127642932,1624751185398704445629757084188,2457871026957756859612862822442301

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

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

%e The a(0) = 1 through a(3) = 29 set-systems:

%e   {}  {}     {}           {}

%e       {{1}}  {{1}}        {{1}}

%e              {{2}}        {{2}}

%e              {{1,2}}      {{3}}

%e              {{1},{1,2}}  {{1,2}}

%e              {{2},{1,2}}  {{1,3}}

%e                           {{2,3}}

%e                           {{1,2,3}}

%e                           {{1},{1,2}}

%e                           {{1},{1,3}}

%e                           {{2},{1,2}}

%e                           {{2},{2,3}}

%e                           {{3},{1,3}}

%e                           {{3},{2,3}}

%e                           {{1},{1,2,3}}

%e                           {{2},{1,2,3}}

%e                           {{3},{1,2,3}}

%e                           {{1,2},{1,2,3}}

%e                           {{1,3},{1,2,3}}

%e                           {{2,3},{1,2,3}}

%e                           {{1},{1,2},{1,2,3}}

%e                           {{1},{1,3},{1,2,3}}

%e                           {{2},{1,2},{1,2,3}}

%e                           {{2},{2,3},{1,2,3}}

%e                           {{3},{1,3},{1,2,3}}

%e                           {{3},{2,3},{1,2,3}}

%e                           {{1},{1,2},{1,3},{1,2,3}}

%e                           {{2},{1,2},{2,3},{1,2,3}}

%e                           {{3},{1,3},{2,3},{1,2,3}}

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

%t (* Second program: *)

%t A006058 = Cases[Import["https://oeis.org/A006058/b006058.txt", "Table"], {_, _}][[All, 2]];

%t a[n_] := Sum[Binomial[n, k] A006058[[k + 1]], {k, 0, n}];

%t a /@ Range[0, 18] (* _Jean-François Alcover_, Jan 01 2020 *)

%Y Binomial transform of A006058 (the covering case).

%Y The case closed under union only is A102896.

%Y The case with {} allowed is A306445.

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

%Y The case closed under intersection only is A326901.

%Y The unlabeled version is A326908.

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

%K nonn

%O 0,2

%A _Gus Wiseman_, Aug 04 2019

%E a(16)-a(18) from A006058 by _Jean-François Alcover_, Jan 01 2020