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%I #31 Aug 12 2019 22:30:57
%S 1,1,2,12,232,19230,16113300,1063117943398,225402329237199496416
%N Number of T_0 set-systems covering n vertices that are closed under intersection.
%C First differs from A182507 at a(5) = 19230, A182507(5) = 12848.
%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 a(n) = A326943(n) - A326944(n).
%F a(n) = Sum_{k = 1..n} s(n,k) * A326901(k - 1) where s = A048994.
%F a(n) = Sum_{k = 1..n} s(n,k) * A326902(k) where s = A048994.
%e The a(0) = 1 through a(3) = 12 set-systems:
%e {} {{1}} {{1},{1,2}} {{1},{1,2},{1,3}}
%e {{2},{1,2}} {{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}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]
%Y The version with empty edges allowed is A326943.
%Y Cf. A003465, A059052, A059201, A319637, A326880, A326881, A326901, A326902, A326905, A326944, A326945.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Aug 11 2019
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