A326867 Number of unlabeled connectedness systems on n vertices.

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

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.

Examples

			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}}
		

Links

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

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.

Extensions

a(5) from Andrew Howroyd, Aug 10 2019

a(6) from Andrew Howroyd, Oct 28 2023