A326877 Number of connectedness systems covering n vertices without singletons.

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

Offset

0,4

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.

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

Example

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}}

Mathematica

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

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.

Sequence in context: A349114 A266920 A072447 * A332138 A225698 A151932

Adjacent sequences: A326874 A326875 A326876 * A326878 A326879 A326880

Keyword

nonn,more

Author

Gus Wiseman, Jul 30 2019

Extensions

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

Status

approved