0,3

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).

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

The a(0) = 1 through a(2) = 4 sets of subsets:

{{}} {{},{1}} {{},{1},{2}}

{{},{1},{1,2}}

{{},{2},{1,2}}

{{},{1},{2},{1,2}}

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}]

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.

Sequence in context: A375442 A229528 A109056 * A155204 A290765 A144992

Adjacent sequences: A326941 A326942 A326943 * A326945 A326946 A326947

nonn,more

Gus Wiseman, Aug 08 2019

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

approved