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

1, 2, 5, 34, 1919, 18660178

Offset

0,2

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).

Links

Table of n, a(n) for n=0..5.

Formula

Partial sums of A319637.

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

Example

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}}

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}]

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.

Sequence in context: A358688 A002665 A192222 * A241586 A000665 A058882

Adjacent sequences: A326943 A326944 A326945 * A326947 A326948 A326949

Keyword

nonn,more

Author

Gus Wiseman, Aug 08 2019

Extensions

a(5) from Max Alekseyev, Oct 11 2023

Status

approved