cp's OEIS Frontend

A326967 Number of sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.

%I A326967 #7 Aug 12 2019 22:31:53
%S A326967 2,4,10,92,38362,4020654364,18438434849260080818,
%T A326967 340282363593610212050791236025945013956,
%U A326967 115792089237316195072053288318104625957065868613454666314675263144628100544274
%N A326967 Number of sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.
%C A326967 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 A326967 a(n) = 2 * A326965(n).
%F A326967 Binomial transform of A326960.
%e A326967 The a(0) = 2 through a(2) = 10 sets of subsets:
%e A326967   {}    {}        {}
%e A326967   {{}}  {{}}      {{}}
%e A326967         {{1}}     {{1}}
%e A326967         {{},{1}}  {{2}}
%e A326967                   {{},{1}}
%e A326967                   {{},{2}}
%e A326967                   {{1},{2}}
%e A326967                   {{},{1},{2}}
%e A326967                   {{1},{2},{1,2}}
%e A326967                   {{},{1},{2},{1,2}}
%t A326967 tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]],Length[#]==1&]==Union@@eds;
%t A326967 Table[Length[Select[Subsets[Subsets[Range[n]]],tmQ[#]&]],{n,0,3}]
%Y A326967 Sets of subsets are A001146.
%Y A326967 The unlabeled version is A326951.
%Y A326967 The covering version is A326960.
%Y A326967 The case without empty edges is A326965.
%Y A326967 Sets of subsets whose dual is a weak antichain are A326969.
%Y A326967 Cf. A059052, A059523, A326941, A326966, A326972, A326976, A326977, A326979.
%K A326967 nonn
%O A326967 0,1
%A A326967 _Gus Wiseman_, Aug 10 2019