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A326944
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Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.
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9
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} Stirling1(n,k)*A326881(k). - _Andrew Howroyd_, Aug 14 2019
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EXAMPLE
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The a(0) = 1 through a(2) = 4 sets of subsets:
{{}} {{},{1}} {{},{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}];
Table[Length[Select[Subsets[Subsets[Range[n]]], MemberQ[#, {}]&&Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}]
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CROSSREFS
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The version not closed under intersection is A059201.
The version where {} is not necessarily an edge is A326943.
Cf. A003181, A003465, A055621, A182507, A245567, A316978, A319564, A326906, A326939, A326941, A326945, A326947.
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KEYWORD
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nonn,more
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AUTHOR
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_Gus Wiseman_, Aug 08 2019
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EXTENSIONS
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a(5)-a(7) from _Andrew Howroyd_, Aug 14 2019
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STATUS
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approved
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