cp's OEIS Frontend

A326968 Number of set-systems on n vertices whose dual is a weak antichain.

%I A326968 #6 Aug 12 2019 22:32:02
%S A326968 1,2,6,56,19446
%N A326968 Number of set-systems on n vertices whose dual is a weak antichain.
%C A326968 A set-system is a finite set of finite nonempty sets. 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.
%F A326968 a(n) = A326969(n)/2.
%F A326968 Binomial transform of A326970.
%e A326968 The a(0) = 1 through a(2) = 6 set-systems:
%e A326968   {}  {}     {}
%e A326968       {{1}}  {{1}}
%e A326968              {{2}}
%e A326968              {{1,2}}
%e A326968              {{1},{2}}
%e A326968              {{1},{2},{1,2}}
%t A326968 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326968 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326968 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]
%Y A326968 The case with strict dual is A326965.
%Y A326968 The BII-numbers of these set-systems are A326966.
%Y A326968 The version with empty edges allowed is A326969.
%Y A326968 The covering case is A326970.
%Y A326968 The unlabeled version is A326971.
%Y A326968 Cf. A058891, A059523, A326940, A326972, A326973, A326975, A326978, A326979.
%K A326968 nonn,more
%O A326968 0,2
%A A326968 _Gus Wiseman_, Aug 10 2019