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

1, 2, 4, 12, 107, 6439, 7726965, 2414519001532, 56130437161079183223017, 286386577668298409107773412840148848120595

Offset: 0

Author

Gus Wiseman, Aug 08 2019

Keywords

• nonn
• more

Comments

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).

Examples

			The a(0) = 1 through a(3) = 12 antichains:
  {}  {}     {}         {}
      {{1}}  {{1}}      {{1}}
             {{2}}      {{2}}
             {{1},{2}}  {{3}}
                        {{1},{2}}
                        {{1},{3}}
                        {{2},{3}}
                        {{1,2},{1,3}}
                        {{1,2},{2,3}}
                        {{1},{2},{3}}
                        {{1,3},{2,3}}
                        {{1,2},{1,3},{2,3}}
		

Crossrefs

Antichains of nonempty sets are A014466.

T_0 set-systems are A326940.

The covering case is A245567.

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

Programs

Mathematica

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],stableQ[#,SubsetQ]&&UnsameQ@@dual[#]&]],{n,0,3}]

Formula

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

Extensions

a(5)-a(8) from Andrew Howroyd, Aug 14 2019

a(9), based on A245567, from Patrick De Causmaecker, Jun 01 2023