A326871 Number of unlabeled connectedness systems covering n vertices.

1, 1, 4, 24, 436, 80460, 1689114556

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 covering if every vertex belongs to some edge.

Links

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

Example

Non-isomorphic representatives of the a(0) = 1 through a(3) = 24 connectedness systems:
  {}  {{1}}  {{1,2}}          {{1,2,3}}
             {{1},{2}}        {{1},{2,3}}
             {{2},{1,2}}      {{1},{2},{3}}
             {{1},{2},{1,2}}  {{3},{1,2,3}}
                              {{1},{3},{2,3}}
                              {{2,3},{1,2,3}}
                              {{2},{3},{1,2,3}}
                              {{1},{2,3},{1,2,3}}
                              {{1},{2},{3},{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 non-covering case without singletons is A072444.

The case without singletons is A326899.

First differences of A326867 (the non-covering case).

Euler transform of A326869 (the connected case).

The labeled case is A326870.

Cf. A072445, A072446, A072447, A193674, A323818, A326866, A326868.

Sequence in context: A167140 A010572 A247737 * A012945 A296398 A166947

Adjacent sequences: A326868 A326869 A326870 * A326872 A326873 A326874

Keyword

nonn,more

Author

Gus Wiseman, Jul 29 2019

Extensions

a(5) from Andrew Howroyd, Aug 10 2019

a(6) from Andrew Howroyd, Oct 28 2023

Status

approved