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%I #8 Aug 12 2019 22:32:09
%S 2,4,12,112,38892
%N Number of sets of subsets of {1..n} whose dual is a weak antichain.
%C 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 a(n) = 2 * A326968(n).
%F a(n) = 2 * Sum_{k = 0..n} binomial(n, k) * A326970(k).
%e The a(0) = 2 through a(2) = 12 sets of subsets:
%e {} {} {}
%e {{}} {{}} {{}}
%e {{1}} {{1}}
%e {{},{1}} {{2}}
%e {{1,2}}
%e {{},{1}}
%e {{},{2}}
%e {{1},{2}}
%e {{},{1,2}}
%e {{},{1},{2}}
%e {{1},{2},{1,2}}
%e {{},{1},{2},{1,2}}
%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t Table[Length[Select[Subsets[Subsets[Range[n]]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]
%Y Sets of subsets whose dual is strict are A326941.
%Y The BII-numbers of set-systems whose dual is a weak antichain are A326966.
%Y Sets of subsets whose dual is a (strict) antichain are A326967.
%Y The case without empty edges is A326968.
%Y Cf. A001146, A059052, A326951, A326970, A326971, A326975, A326978.
%K nonn,more
%O 0,1
%A _Gus Wiseman_, Aug 10 2019
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