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A327039
Number of set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).
16
1, 2, 7, 88, 25421, 2077323118, 9221293242272922067, 170141182628636920942528022609657505092
OFFSET
0,2
COMMENTS
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.
FORMULA
Binomial transform of A327040.
EXAMPLE
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}}
MATHEMATICA
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}]
CROSSREFS
The unlabeled multiset partition version is A319752.
The BII-numbers of these set-systems are A326853.
The pairwise intersecting case is A327038.
The covering case is A327040.
The case where the dual is strict is A327052.
Sequence in context: A054919 A119157 A079701 * A096208 A123995 A350754
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 17 2019
EXTENSIONS
a(5)-a(7) from Christian Sievers, Oct 22 2023
STATUS
approved