A326939 Number of T_0 sets of subsets of {1..n} that cover all n vertices.

2, 2, 8, 192, 63384, 4294003272, 18446743983526539408, 340282366920938462946865774750753349904, 115792089237316195423570985008687907841019819456486779364848020385134373080448

Offset: 0

Author

Gus Wiseman, Aug 07 2019

Keywords

• nonn

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, counted with multiplicity. 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) = 8 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}
		

Crossrefs

The non-T_0 version is A000371.

The case without empty edges is A059201.

The non-covering version is A326941.

The unlabeled version is A326942.

The case closed under intersection is A326943.

Cf. A003180, A003181, A003465, A316978, A319564, A319637, A326940, A326947.

Programs

Mathematica

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

Formula

a(n) = 2 * A059201(n).

Inverse binomial transform of A326941.