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Number of T_0 antichains of nonempty subsets of {1..n}.
7

%I #20 Jun 02 2023 01:13:48

%S 1,2,4,12,107,6439,7726965,2414519001532,56130437161079183223017,

%T 286386577668298409107773412840148848120595

%N Number of T_0 antichains of nonempty subsets of {1..n}.

%C 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 Binomial transform of A245567, if we assume A245567(0) = 1.

%e The a(0) = 1 through a(3) = 12 antichains:

%e {} {} {} {}

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

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

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

%e {{1},{2}}

%e {{1},{3}}

%e {{2},{3}}

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

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

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

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

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

%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[#,SubsetQ]&&UnsameQ@@dual[#]&]],{n,0,3}]

%Y Antichains of nonempty sets are A014466.

%Y T_0 set-systems are A326940.

%Y The covering case is A245567.

%Y Cf. A006126, A059201, A059052, A245567, A319559, A319564, A326030, A326946, A326947.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Aug 08 2019

%E a(5)-a(8) from _Andrew Howroyd_, Aug 14 2019

%E a(9), based on A245567, from _Patrick De Causmaecker_, Jun 01 2023