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1, 1, 4, 72, 25104, 2077196832, 9221293229809363008, 170141182628636920877978969957369949312

a(5)-a(7) from Christian Sievers, Oct 22 2023

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The non-covering case version is A327039.

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. This sequence counts covering set-systems that are cointersecting, meaning their dual is pairwise intersecting.

allocated for Gus WisemanNumber of set-systems covering n vertices, every two of which appear together in some edge (cointersecting).

1, 1, 4, 72, 25104

0,3

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. This sequence counts covering set-systems that are cointersecting, meaning their dual is pairwise intersecting.

Inverse binomial transform of A327039.

The a(0) = 1 through a(2) = 4 set-systems:

{} {{1}} {{1,2}}

{{1},{1,2}}

{{2},{1,2}}

{{1},{2},{1,2}}

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];

stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];

Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&stableQ[dual[#], Intersection[#1, #2]=={}&]&]], {n, 0, 3}]

The unlabeled multiset partition version is A319752.

The BII-numbers of these set-systems are A326853.

The antichain case is A327020.

The pairwise intersecting case is A327037.

The non-covering case is A327039.

The case where the dual is strict is A327053.

Cf. A003465, A305843, A305844, A306006, A319774, A327052.

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nonn,more

Gus Wiseman, Aug 18 2019

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allocated for Gus Wiseman

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