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Number of set-systems on n vertices whose dual is a weak antichain.
12

%I #6 Aug 12 2019 22:32:02

%S 1,2,6,56,19446

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

%C 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 a(n) = A326969(n)/2.

%F Binomial transform of A326970.

%e The a(0) = 1 through a(2) = 6 set-systems:

%e {} {} {}

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

%e {{2}}

%e {{1,2}}

%e {{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],{1,n}]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

%Y The case with strict dual is A326965.

%Y The BII-numbers of these set-systems are A326966.

%Y The version with empty edges allowed is A326969.

%Y The covering case is A326970.

%Y The unlabeled version is A326971.

%Y Cf. A058891, A059523, A326940, A326972, A326973, A326975, A326978, A326979.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Aug 10 2019