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A326959 Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection. 5

%I

%S 1,2,5,22,297,20536,16232437,1063231148918,225402337742595309857

%N Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection.

%C 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).

%F Binomial transform of A309615.

%e The a(0) = 1 through a(3) = 22 set-systems:

%e {} {} {} {}

%e {{1}} {{1}} {{1}}

%e {{2}} {{2}}

%e {{1},{1,2}} {{3}}

%e {{2},{1,2}} {{1},{1,2}}

%e {{1},{1,3}}

%e {{2},{1,2}}

%e {{2},{2,3}}

%e {{3},{1,3}}

%e {{3},{2,3}}

%e {{1},{1,2},{1,3}}

%e {{2},{1,2},{2,3}}

%e {{3},{1,3},{2,3}}

%e {{1},{1,2},{1,2,3}}

%e {{1},{1,3},{1,2,3}}

%e {{2},{1,2},{1,2,3}}

%e {{2},{2,3},{1,2,3}}

%e {{3},{1,3},{1,2,3}}

%e {{3},{2,3},{1,2,3}}

%e {{1},{1,2},{1,3},{1,2,3}}

%e {{2},{1,2},{2,3},{1,2,3}}

%e {{3},{1,3},{2,3},{1,2,3}}

%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];

%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

%Y The covering case is A309615.

%Y T_0 set-systems are A326940.

%Y The version with empty edges allowed is A326945.

%Y Cf. A051185, A058891, A059201, A316978, A319559, A309615, A319637, A326943, A326944, A326946, A326947, A326959.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Aug 13 2019

%E a(5)-a(8) from _Andrew Howroyd_, Aug 14 2019

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Last modified February 18 00:37 EST 2020. Contains 332006 sequences. (Running on oeis4.)