A326870 Number of connectedness systems covering n vertices.

1, 1, 5, 77, 6377, 8097721, 1196051135917

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 covering if every vertex belongs to some edge.

Examples

			The a(2) = 5 connectedness systems:
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}
		

Links

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

Crossrefs

Inverse binomial transform of A326866 (the non-covering case).

Exponential transform of A326868 (the connected case).

The unlabeled case is A326871.

The BII-numbers of these set-systems are A326872.

The case without singletons is A326877.

Cf. A072445, A072446, A072447, A102896, A323818, A326867, A326869.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&SubsetQ[#,Union@@@Select[Tuples[#,2],Intersection@@#!={}&]]&]],{n,0,4}]

Extensions

a(6) corrected by Christian Sievers, Oct 28 2023