cp's OEIS Frontend

A326901 Number of set-systems (without {}) on n vertices that are closed under intersection.

%I A326901 #16 Aug 11 2019 13:50:49
%S A326901 1,2,6,32,418,23702,16554476,1063574497050,225402367516942398102
%N A326901 Number of set-systems (without {}) on n vertices that are closed under intersection.
%C A326901 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 A326901 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 A326901 a(n) = 1 + Sum_{k=0, n-1} binomial(n,k)*A102895(k). - _Andrew Howroyd_, Aug 10 2019
%e A326901 The a(3) = 32 set-systems:
%e A326901   {}  {{1}}    {{1}{12}}    {{1}{12}{13}}   {{1}{12}{13}{123}}
%e A326901       {{2}}    {{1}{13}}    {{2}{12}{23}}   {{2}{12}{23}{123}}
%e A326901       {{3}}    {{2}{12}}    {{3}{13}{23}}   {{3}{13}{23}{123}}
%e A326901       {{12}}   {{2}{23}}    {{1}{12}{123}}
%e A326901       {{13}}   {{3}{13}}    {{1}{13}{123}}
%e A326901       {{23}}   {{3}{23}}    {{2}{12}{123}}
%e A326901       {{123}}  {{1}{123}}   {{2}{23}{123}}
%e A326901                {{2}{123}}   {{3}{13}{123}}
%e A326901                {{3}{123}}   {{3}{23}{123}}
%e A326901                {{12}{123}}
%e A326901                {{13}{123}}
%e A326901                {{23}{123}}
%t A326901 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]
%Y A326901 The case with union instead of intersection is A102896.
%Y A326901 The case closed under union and intersection is A326900.
%Y A326901 The covering case is A326902.
%Y A326901 The connected case is A326903.
%Y A326901 The unlabeled version is A326904.
%Y A326901 The BII-numbers of these set-systems are A326905.
%Y A326901 Cf. A000798, A001930, A006058, A102895, A102898, A182507, A326866, A326876, A326878, A326882.
%K A326901 nonn,more
%O A326901 0,2
%A A326901 _Gus Wiseman_, Aug 04 2019
%E A326901 a(5)-a(8) from _Andrew Howroyd_, Aug 10 2019