A326867 Number of unlabeled connectedness systems on n vertices.

1, 2, 6, 30, 466, 80926, 1689195482

Offset

0,2

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.

Links

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

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Example

Non-isomorphic representatives of the a(0) = 1 through a(3) = 30 connectedness systems:
  {}  {}     {}               {}
      {{1}}  {{1}}            {{1}}
             {{1,2}}          {{1,2}}
             {{1},{2}}        {{1},{2}}
             {{2},{1,2}}      {{1,2,3}}
             {{1},{2},{1,2}}  {{1},{2,3}}
                              {{2},{1,2}}
                              {{1},{2},{3}}
                              {{3},{1,2,3}}
                              {{1},{2},{1,2}}
                              {{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 case without singletons is A072444.

The labeled case is A326866.

The connected case is A326869.

Partial sums of A326871 (the covering case).

Cf. A072445, A072446, A072447, A102896, A306445, A326870, A326872.

Sequence in context: A076926 A092023 A112723 * A320830 A074777 A007280

Adjacent sequences: A326864 A326865 A326866 * A326868 A326869 A326870

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