A326946 Number of unlabeled T_0 set-systems on n vertices.

1, 2, 5, 34, 1919, 18660178

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

			Non-isomorphic representatives of the a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{1},{2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}
		

Crossrefs

The non-T_0 version is A000612.

The antichain case is A245567.

The covering case is A319637.

The labeled version is A326940.

The version with empty edges allowed is A326949.

Cf. A003180, A055621, A059052, A059201, A316978, A319559, A319564, A326942.

Programs

Mathematica

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Union[normclut/@Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&]]],{n,0,3}]

Formula

Partial sums of A319637.

a(n) = A326949(n)/2.

Extensions

a(5) from Max Alekseyev, Oct 11 2023