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A326959
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Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection.
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5
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OFFSET
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0,2
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COMMENTS
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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).
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LINKS
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FORMULA
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EXAMPLE
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The a(0) = 1 through a(3) = 22 set-systems:
{} {} {} {}
{{1}} {{1}} {{1}}
{{2}} {{2}}
{{1},{1,2}} {{3}}
{{2},{1,2}} {{1},{1,2}}
{{1},{1,3}}
{{2},{1,2}}
{{2},{2,3}}
{{3},{1,3}}
{{3},{2,3}}
{{1},{1,2},{1,3}}
{{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}}
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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], {1, n}]], UnsameQ@@dual[#]&&SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}]
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CROSSREFS
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The version with empty edges allowed is A326945.
Cf. A051185, A058891, A059201, A316978, A319559, A309615, A319637, A326943, A326944, A326946, A326947, A326959.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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