%I #7 Aug 19 2019 08:50:51

%S 1,2,5,16,81,2595

%N Number of antichains of distinct sets covering a subset of {1..n} whose dual is a weak antichain.

%C A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. 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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

%e The a(0) = 1 through a(3) = 16 antichains:

%e {} {} {} {}

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

%e {{2}} {{2}}

%e {{1,2}} {{3}}

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

%e {{1,3}}

%e {{2,3}}

%e {{1},{2}}

%e {{1,2,3}}

%e {{1},{3}}

%e {{2},{3}}

%e {{1},{2,3}}

%e {{2},{1,3}}

%e {{3},{1,2}}

%e {{1},{2},{3}}

%e {{1,2},{1,3},{2,3}}

%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];

%t stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

%Y Antichains are A000372.

%Y The covering case is A319639.

%Y The non-isomorphic multiset partition version is A319721.

%Y The BII-numbers of these set-systems are the intersection of A326910 and A326853.

%Y Set-systems whose dual is a weak antichain are A326968.

%Y The unlabeled version is A327018.

%Y Cf. A006126, A318099, A293606, A326704, A326950, A327020, A327057.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Aug 18 2019