cp's OEIS Frontend

A327062 Number of antichains of distinct sets covering a subset of {1..n} whose dual is a weak antichain.

%I A327062 #7 Aug 19 2019 08:50:51
%S A327062 1,2,5,16,81,2595
%N A327062 Number of antichains of distinct sets covering a subset of {1..n} whose dual is a weak antichain.
%C A327062 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 A327062 The a(0) = 1 through a(3) = 16 antichains:
%e A327062   {}  {}     {}         {}
%e A327062       {{1}}  {{1}}      {{1}}
%e A327062              {{2}}      {{2}}
%e A327062              {{1,2}}    {{3}}
%e A327062              {{1},{2}}  {{1,2}}
%e A327062                         {{1,3}}
%e A327062                         {{2,3}}
%e A327062                         {{1},{2}}
%e A327062                         {{1,2,3}}
%e A327062                         {{1},{3}}
%e A327062                         {{2},{3}}
%e A327062                         {{1},{2,3}}
%e A327062                         {{2},{1,3}}
%e A327062                         {{3},{1,2}}
%e A327062                         {{1},{2},{3}}
%e A327062                         {{1,2},{1,3},{2,3}}
%t A327062 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A327062 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 A327062 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A327062 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],stableQ[dual[#],SubsetQ]&]],{n,0,3}]
%Y A327062 Antichains are A000372.
%Y A327062 The covering case is A319639.
%Y A327062 The non-isomorphic multiset partition version is A319721.
%Y A327062 The BII-numbers of these set-systems are the intersection of A326910 and A326853.
%Y A327062 Set-systems whose dual is a weak antichain are A326968.
%Y A327062 The unlabeled version is A327018.
%Y A327062 Cf. A006126, A318099, A293606, A326704, A326950, A327020, A327057.
%K A327062 nonn,more
%O A327062 0,2
%A A327062 _Gus Wiseman_, Aug 18 2019