A309615 Number of T_0 set-systems covering n vertices that are closed under intersection.

1, 1, 2, 12, 232, 19230, 16113300, 1063117943398, 225402329237199496416

Offset

0,3

Comments

First differs from A182507 at a(5) = 19230, A182507(5) = 12848.

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}}. The T_0 condition means that the dual is strict (no repeated edges).

Links

Table of n, a(n) for n=0..8.

Formula

a(n) = A326943(n) - A326944(n).

a(n) = Sum_{k = 1..n} s(n,k) * A326901(k - 1) where s = A048994.

a(n) = Sum_{k = 1..n} s(n,k) * A326902(k) where s = A048994.

Example

The a(0) = 1 through a(3) = 12 set-systems:
  {}  {{1}}  {{1},{1,2}}  {{1},{1,2},{1,3}}
             {{2},{1,2}}  {{2},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1},{1,2},{1,2,3}}
                          {{1},{1,3},{1,2,3}}
                          {{2},{1,2},{1,2,3}}
                          {{2},{2,3},{1,2,3}}
                          {{3},{1,3},{1,2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{1},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

Mathematica

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}]

Crossrefs

The version with empty edges allowed is A326943.

Cf. A003465, A059052, A059201, A319637, A326880, A326881, A326901, A326902, A326905, A326944, A326945.

Sequence in context: A011807 A182507 A348877 * A166316 A011840 A296462

Adjacent sequences: A309612 A309613 A309614 * A309616 A309617 A309618

Keyword

nonn,more

Author

Gus Wiseman, Aug 11 2019

Status

approved