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A327040 Number of set-systems covering n vertices, every two of which appear together in some edge (cointersecting). 16
1, 1, 4, 72, 25104, 2077196832, 9221293229809363008, 170141182628636920877978969957369949312 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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 covering set-systems that are cointersecting, meaning their dual is pairwise intersecting.
LINKS
FORMULA
Inverse binomial transform of A327039.
EXAMPLE
The a(0) = 1 through a(2) = 4 set-systems:
{} {{1}} {{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}]], Union@@#==Range[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 antichain case is A327020.
The pairwise intersecting case is A327037.
The non-covering version is A327039.
The case where the dual is strict is A327053.
Sequence in context: A172478 A087315 A081460 * A327112 A284673 A055556
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 18 2019
EXTENSIONS
a(5)-a(7) from Christian Sievers, Oct 22 2023
STATUS
approved

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Last modified July 12 19:17 EDT 2024. Contains 374252 sequences. (Running on oeis4.)