A326877 Number of connectedness systems covering n vertices without singletons.

1, 0, 1, 8, 381, 252080, 18687541309

Offset: 0

Author

Gus Wiseman, Jul 30 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(3) = 8 covering connectedness systems without singletons:
  {{1,2,3}}
  {{1,2},{1,2,3}}
  {{1,3},{1,2,3}}
  {{2,3},{1,2,3}}
  {{1,2},{1,3},{1,2,3}}
  {{1,2},{2,3},{1,2,3}}
  {{1,3},{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

Inverse binomial transform of A072446 (the non-covering case).

Exponential transform of A072447 if we assume A072447(1) = 0 (the connected case).

The case with singletons is A326870.

The BII-numbers of these set-systems are A326873.

Cf. A072444, A072445, A102896, A323818, A326866, A326867, A326868, A326871, A326872.

Programs

Mathematica

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

Extensions

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