A326870 Number of connectedness systems covering n vertices.

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

Offset

0,3

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

Mathematica

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

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.

Sequence in context: A009485 A188455 A015056 * A327903 A214443 A015973

Adjacent sequences: A326867 A326868 A326869 * A326871 A326872 A326873

Keyword

nonn,more

Author

Gus Wiseman, Jul 29 2019

Extensions

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

Status

approved