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%I #8 Aug 12 2019 22:31:45
%S 1,2,5,46,19181,2010327182,9219217424630040409,
%T 170141181796805106025395618012972506978,
%U 57896044618658097536026644159052312978532934306727333157337631572314050272137
%N Number of set-systems on n vertices where every covered vertex is the unique common element of some subset of the edges.
%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}}. An antichain is a set-system where no edge is a subset of any other. This sequence counts set-systems whose dual is a (strict) antichain, also called T_1 set-systems.
%F Binomial transform of A326961.
%F a(n) = A326967(n)/2.
%e The a(0) = 1 through a(2) = 5 set-systems:
%e {} {} {}
%e {{1}} {{1}}
%e {{2}}
%e {{1},{2}}
%e {{1},{2},{1,2}}
%t tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]],Length[#]==1&]==Union@@eds;
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],tmQ]],{n,0,3}]
%Y Set-systems are A058891.
%Y T_0 set-systems are A326940.
%Y The covering case is A326961.
%Y The version with empty edges allowed is A326967.
%Y Set-systems whose dual is a weak antichain are A326968.
%Y The unlabeled version is A326972.
%Y The BII_numbers of these set-systems are A326979.
%Y Cf. A059052, A326951, A326966, A326970, A326971, A326976, A326977.
%K nonn
%O 0,2
%A _Gus Wiseman_, Aug 10 2019
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