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A327039
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Number of set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).
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16
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OFFSET
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0,2
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COMMENTS
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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 set-systems that are cointersecting, meaning their dual is pairwise intersecting.
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LINKS
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FORMULA
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EXAMPLE
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The a(0) = 1 through a(2) = 7 set-systems:
{} {} {}
{{1}} {{1}}
{{2}}
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}
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MATHEMATICA
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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}]], stableQ[dual[#], Intersection[#1, #2]=={}&]&]], {n, 0, 3}]
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CROSSREFS
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The unlabeled multiset partition version is A319752.
The BII-numbers of these set-systems are A326853.
The pairwise intersecting case is A327038.
The case where the dual is strict is A327052.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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