A326869 Number of unlabeled connected connectedness systems on n vertices.

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

Offset: 0

Author

Gus Wiseman, Jul 29 2019

Keywords

• nonn
• more

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.

Examples

			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.

Extensions

a(5) from Andrew Howroyd, Aug 16 2019

a(6) from Andrew Howroyd, Oct 28 2023