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A326944
Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.
9
1, 1, 4, 58, 3846, 2685550, 151873991914, 28175291154649937052
OFFSET
0,3
COMMENTS
The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks 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) = Sum_{k=0..n} Stirling1(n,k)*A326881(k). - Andrew Howroyd, Aug 14 2019
EXAMPLE
The a(0) = 1 through a(2) = 4 sets of subsets:
{{}} {{},{1}} {{},{1},{2}}
{{},{1},{1,2}}
{{},{2},{1,2}}
{{},{1},{2},{1,2}}
MATHEMATICA
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]], MemberQ[#, {}]&&Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}]
CROSSREFS
The version not closed under intersection is A059201.
The non-T_0 version is A326881.
The version where {} is not necessarily an edge is A326943.
Sequence in context: A375442 A229528 A109056 * A155204 A290765 A144992
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 08 2019
EXTENSIONS
a(5)-a(7) from Andrew Howroyd, Aug 14 2019
STATUS
approved