A326869 Number of unlabeled connected connectedness systems on n vertices.

1, 1, 3, 20, 406, 79964, 1689032658

Offset

0,3

Comments

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges. It is connected if it contains an edge with all the vertices.

Links

Table of n, a(n) for n=0..6.

Example

Non-isomorphic representatives of the a(3) = 20 connected connectedness systems:
  {{1,2,3}}
  {{3},{1,2,3}}
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{1},{3},{2,3},{1,2,3}}
  {{2},{3},{2,3},{1,2,3}}
  {{2},{1,3},{2,3},{1,2,3}}
  {{3},{1,3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{2,3},{1,2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{2},{3},{1,3},{2,3},{1,2,3}}
  {{3},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,3},{2,3},{1,2,3}}
  {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

The case without singletons is A072445.

Connected set-systems are A092918.

The not necessarily connected case is A326867.

The labeled case is A326868.

Euler transform is A326871 (the covering case).

Cf. A072444, A072446, A072447, A102896, A323818, A326866, A326870, A326879.

Sequence in context: A201824 A203519 A003150 * A203194 A322455 A138897

Adjacent sequences: A326866 A326867 A326868 * A326870 A326871 A326872

Keyword

nonn,more

Author

Gus Wiseman, Jul 29 2019

Extensions

a(5) from Andrew Howroyd, Aug 16 2019

a(6) from Andrew Howroyd, Oct 28 2023

Status

approved