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A309615
Number of T_0 set-systems covering n vertices that are closed under intersection.
5
1, 1, 2, 12, 232, 19230, 16113300, 1063117943398, 225402329237199496416
OFFSET
0,3
COMMENTS
First differs from A182507 at a(5) = 19230, A182507(5) = 12848.
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}}. The T_0 condition means that the dual is strict (no repeated edges).
FORMULA
a(n) = A326943(n) - A326944(n).
a(n) = Sum_{k = 1..n} s(n,k) * A326901(k - 1) where s = A048994.
a(n) = Sum_{k = 1..n} s(n,k) * A326902(k) where s = A048994.
EXAMPLE
The a(0) = 1 through a(3) = 12 set-systems:
{} {{1}} {{1},{1,2}} {{1},{1,2},{1,3}}
{{2},{1,2}} {{2},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1},{1,2},{1,2,3}}
{{1},{1,3},{1,2,3}}
{{2},{1,2},{1,2,3}}
{{2},{2,3},{1,2,3}}
{{3},{1,3},{1,2,3}}
{{3},{2,3},{1,2,3}}
{{1},{1,2},{1,3},{1,2,3}}
{{2},{1,2},{2,3},{1,2,3}}
{{3},{1,3},{2,3},{1,2,3}}
MATHEMATICA
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}]
CROSSREFS
The version with empty edges allowed is A326943.
Sequence in context: A011807 A182507 A348877 * A166316 A011840 A296462
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 11 2019
STATUS
approved