cp's OEIS Frontend

A326903 Number of set-systems (without {}) on n vertices that are closed under intersection and have an edge containing all of the vertices, or Moore families without {}.

%I A326903 #7 Aug 11 2019 14:37:55
%S A326903 0,1,3,16,209,11851,8277238,531787248525,112701183758471199051
%N A326903 Number of set-systems (without {}) on n vertices that are closed under intersection and have an edge containing all of the vertices, or Moore families without {}.
%C A326903 A set-system is a finite set of finite nonempty sets, so no two edges of such a set-system can be disjoint.
%C A326903 If {} is allowed, we get Moore families (A102896, cf A102895).
%H A326903 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 A326903 a(n) = A326901(n) / 2 for n > 0. - _Andrew Howroyd_, Aug 10 2019
%e A326903 The a(1) = 1 through a(3) = 16 set-systems:
%e A326903   {{1}}  {{1,2}}      {{1,2,3}}
%e A326903          {{1},{1,2}}  {{1},{1,2,3}}
%e A326903          {{2},{1,2}}  {{2},{1,2,3}}
%e A326903                       {{3},{1,2,3}}
%e A326903                       {{1,2},{1,2,3}}
%e A326903                       {{1,3},{1,2,3}}
%e A326903                       {{2,3},{1,2,3}}
%e A326903                       {{1},{1,2},{1,2,3}}
%e A326903                       {{1},{1,3},{1,2,3}}
%e A326903                       {{2},{1,2},{1,2,3}}
%e A326903                       {{2},{2,3},{1,2,3}}
%e A326903                       {{3},{1,3},{1,2,3}}
%e A326903                       {{3},{2,3},{1,2,3}}
%e A326903                       {{1},{1,2},{1,3},{1,2,3}}
%e A326903                       {{2},{1,2},{2,3},{1,2,3}}
%e A326903                       {{3},{1,3},{2,3},{1,2,3}}
%t A326903 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],MemberQ[#,Range[n]]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]
%Y A326903 The case closed under union and intersection is A006058.
%Y A326903 The case with union instead of intersection is A102894.
%Y A326903 The unlabeled version is A193674.
%Y A326903 The case without requiring the maximum edge is A326901.
%Y A326903 The covering case is A326902.
%Y A326903 Cf. A000798, A001930, A102895, A102898, A326866, A326876, A326878, A326882, A326904.
%K A326903 nonn,more
%O A326903 0,3
%A A326903 _Gus Wiseman_, Aug 04 2019
%E A326903 a(5)-a(8) from _Andrew Howroyd_, Aug 10 2019