A326901 - OEIS

%I #16 Aug 11 2019 13:50:49

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

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

%C 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.

%H M. Habib and L. Nourine, <a href="https://doi.org/10.1016/j.disc.2004.11.010">The number of Moore families on n = 6</a>, Discrete Math., 294 (2005), 291-296.

%F a(n) = 1 + Sum_{k=0, n-1} binomial(n,k)*A102895(k). - _Andrew Howroyd_, Aug 10 2019

%e The a(3) = 32 set-systems:

%e   {}  {{1}}    {{1}{12}}    {{1}{12}{13}}   {{1}{12}{13}{123}}

%e       {{2}}    {{1}{13}}    {{2}{12}{23}}   {{2}{12}{23}{123}}

%e       {{3}}    {{2}{12}}    {{3}{13}{23}}   {{3}{13}{23}{123}}

%e       {{12}}   {{2}{23}}    {{1}{12}{123}}

%e       {{13}}   {{3}{13}}    {{1}{13}{123}}

%e       {{23}}   {{3}{23}}    {{2}{12}{123}}

%e       {{123}}  {{1}{123}}   {{2}{23}{123}}

%e                {{2}{123}}   {{3}{13}{123}}

%e                {{3}{123}}   {{3}{23}{123}}

%e                {{12}{123}}

%e                {{13}{123}}

%e                {{23}{123}}

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

%Y The case with union instead of intersection is A102896.

%Y The case closed under union and intersection is A326900.

%Y The covering case is A326902.

%Y The connected case is A326903.

%Y The unlabeled version is A326904.

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

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

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Aug 04 2019

%E a(5)-a(8) from _Andrew Howroyd_, Aug 10 2019