A326940 Number of T_0 set-systems on n vertices.

1, 2, 7, 112, 32105, 2147161102, 9223372004645756887, 170141183460469231537996491362807709908, 57896044618658097711785492504343953921871039195927143534469727707459805807105

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, 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).

Links

Table of n, a(n) for n=0..8.

Formula

Binomial transform of A059201.

Example

The a(0) = 1 through a(2) = 7 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

Mathematica

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

Crossrefs

The non-T_0 version is A058891 shifted to the left.

The covering case is A059201.

The version with empty edges is A326941.

The unlabeled version is A326946.

Cf. A003180, A316978, A319559, A319564, A319637, A326939, A326947, A326949.

Sequence in context: A224445 A000157 A264999 * A326964 A034902 A101429

Adjacent sequences: A326937 A326938 A326939 * A326941 A326942 A326943

Keyword

nonn

Author

Gus Wiseman, Aug 07 2019

Status

approved