cp's OEIS Frontend

A326950 Number of T_0 antichains of nonempty subsets of {1..n}.

%I A326950 #20 Jun 02 2023 01:13:48
%S A326950 1,2,4,12,107,6439,7726965,2414519001532,56130437161079183223017,
%T A326950 286386577668298409107773412840148848120595
%N A326950 Number of T_0 antichains of nonempty subsets of {1..n}.
%C A326950 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}}. The T_0 condition means that the dual is strict (no repeated edges).
%F A326950 Binomial transform of A245567, if we assume A245567(0) = 1.
%e A326950 The a(0) = 1 through a(3) = 12 antichains:
%e A326950   {}  {}     {}         {}
%e A326950       {{1}}  {{1}}      {{1}}
%e A326950              {{2}}      {{2}}
%e A326950              {{1},{2}}  {{3}}
%e A326950                         {{1},{2}}
%e A326950                         {{1},{3}}
%e A326950                         {{2},{3}}
%e A326950                         {{1,2},{1,3}}
%e A326950                         {{1,2},{2,3}}
%e A326950                         {{1},{2},{3}}
%e A326950                         {{1,3},{2,3}}
%e A326950                         {{1,2},{1,3},{2,3}}
%t A326950 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326950 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326950 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],stableQ[#,SubsetQ]&&UnsameQ@@dual[#]&]],{n,0,3}]
%Y A326950 Antichains of nonempty sets are A014466.
%Y A326950 T_0 set-systems are A326940.
%Y A326950 The covering case is A245567.
%Y A326950 Cf. A006126, A059201, A059052, A245567, A319559, A319564, A326030, A326946, A326947.
%K A326950 nonn,more
%O A326950 0,2
%A A326950 _Gus Wiseman_, Aug 08 2019
%E A326950 a(5)-a(8) from _Andrew Howroyd_, Aug 14 2019
%E A326950 a(9), based on A245567, from _Patrick De Causmaecker_, Jun 01 2023