A326944 Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.

1, 1, 4, 58, 3846, 2685550, 151873991914, 28175291154649937052

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) = 1 through a(2) = 4 sets of subsets:
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}
		

Crossrefs

The version not closed under intersection is A059201.

The non-T_0 version is A326881.

The version where {} is not necessarily an edge is A326943.

Cf. A003181, A003465, A055621, A182507, A245567, A316978, A319564, A326906, A326939, A326941, A326945, A326947.

Programs

Mathematica

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

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A326881(k). - Andrew Howroyd, Aug 14 2019

Extensions

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