A326941 Number of T_0 sets of subsets of {1..n}.

2, 4, 14, 224, 64210, 4294322204, 18446744009291513774, 340282366920938463075992982725615419816, 115792089237316195423570985008687907843742078391854287068939455414919611614210

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) = 14 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{2}}
                  {{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 A001146.

The covering case is A326939.

The case without empty edges is A326940.

The unlabeled version is A326949.

Cf. A003180, A059201, A316978, A319559, A319564, A319637, A326946, A326947.

Programs

Mathematica

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

Formula

a(n) = 2 * A326940(n).

Binomial transform of A326939.

Extensions

a(5)-a(8) from Andrew Howroyd, Aug 14 2019