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

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

Offset

0,2

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).

Links

Table of n, a(n) for n=0..9.

Formula

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

Example

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}}

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}]

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.

Sequence in context: A325502 A038791 A327563 * A001696 A276534 A326969

Adjacent sequences: A326947 A326948 A326949 * A326951 A326952 A326953

Keyword

nonn,more

Author

Gus Wiseman, Aug 08 2019

Extensions

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

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

Status

approved