A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

2, 4, 12, 96, 4404, 2725942, 151906396568, 28175293281055562650

Offset: 0

Author

Gus Wiseman, Aug 08 2019

Keywords

• nonn
• more

Comments

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks 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) = 2 through a(2) = 12 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}
		

Crossrefs

The non-T_0 version is A102897.

The version not closed under intersection is A326941.

The covering case is A326943.

The case without empty edges is A326959.

Cf. A003180, A182507, A316978, A319564, A326906, A326939, A326940, A326944, A326947.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n]]],UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Formula

Binomial transform of A326943.

Extensions

a(5)-a(7) from Andrew Howroyd, Aug 14 2019