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A326969 Number of sets of subsets of {1..n} whose dual is a weak antichain.

%I A326969 #8 Aug 12 2019 22:32:09
%S A326969 2,4,12,112,38892
%N A326969 Number of sets of subsets of {1..n} whose dual is a weak antichain.
%C A326969 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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.
%F A326969 a(n) = 2 * A326968(n).
%F A326969 a(n) = 2 * Sum_{k = 0..n} binomial(n, k) * A326970(k).
%e A326969 The a(0) = 2 through a(2) = 12 sets of subsets:
%e A326969   {}    {}        {}
%e A326969   {{}}  {{}}      {{}}
%e A326969         {{1}}     {{1}}
%e A326969         {{},{1}}  {{2}}
%e A326969                   {{1,2}}
%e A326969                   {{},{1}}
%e A326969                   {{},{2}}
%e A326969                   {{1},{2}}
%e A326969                   {{},{1,2}}
%e A326969                   {{},{1},{2}}
%e A326969                   {{1},{2},{1,2}}
%e A326969                   {{},{1},{2},{1,2}}
%t A326969 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326969 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326969 Table[Length[Select[Subsets[Subsets[Range[n]]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]
%Y A326969 Sets of subsets whose dual is strict are A326941.
%Y A326969 The BII-numbers of set-systems whose dual is a weak antichain are A326966.
%Y A326969 Sets of subsets whose dual is a (strict) antichain are A326967.
%Y A326969 The case without empty edges is A326968.
%Y A326969 Cf. A001146, A059052, A326951, A326970, A326971, A326975, A326978.
%K A326969 nonn,more
%O A326969 0,1
%A A326969 _Gus Wiseman_, Aug 10 2019