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Number of sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.
5

%I #7 Aug 12 2019 22:31:53

%S 2,4,10,92,38362,4020654364,18438434849260080818,

%T 340282363593610212050791236025945013956,

%U 115792089237316195072053288318104625957065868613454666314675263144628100544274

%N Number of sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.

%C Alternatively, these are sets of subsets of {1..n} whose dual is a (strict) antichain, also called T_1 sets of subsets. The dual of a set of subsets 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 of sets, none of which is a subset of any other.

%F a(n) = 2 * A326965(n).

%F Binomial transform of A326960.

%e The a(0) = 2 through a(2) = 10 sets of subsets:

%e {} {} {}

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

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

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

%e {{},{1}}

%e {{},{2}}

%e {{1},{2}}

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

%e {{1},{2},{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]]],tmQ[#]&]],{n,0,3}]

%Y Sets of subsets are A001146.

%Y The unlabeled version is A326951.

%Y The covering version is A326960.

%Y The case without empty edges is A326965.

%Y Sets of subsets whose dual is a weak antichain are A326969.

%Y Cf. A059052, A059523, A326941, A326966, A326972, A326976, A326977, A326979.

%K nonn

%O 0,1

%A _Gus Wiseman_, Aug 10 2019