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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 {}.
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%I #7 Aug 11 2019 14:37:55

%S 0,1,3,16,209,11851,8277238,531787248525,112701183758471199051

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

%C If {} is allowed, we get Moore families (A102896, cf A102895).

%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) = A326901(n) / 2 for n > 0. - _Andrew Howroyd_, Aug 10 2019

%e The a(1) = 1 through a(3) = 16 set-systems:

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

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

%e {{2},{1,2}} {{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}]],MemberQ[#,Range[n]]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

%Y The case closed under union and intersection is A006058.

%Y The case with union instead of intersection is A102894.

%Y The unlabeled version is A193674.

%Y The case without requiring the maximum edge is A326901.

%Y The covering case is A326902.

%Y Cf. A000798, A001930, A102895, A102898, A326866, A326876, A326878, A326882, A326904.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Aug 04 2019

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