

A326866


Number of connectedness systems on n vertices.


26




OFFSET

0,2


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 two overlapping edges.


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.


FORMULA

a(n) = 2^n * A072446(n).


EXAMPLE

The a(0) = 1 through a(2) = 8 connectedness systems:
{} {} {}
{{1}} {{1}}
{{2}}
{{1,2}}
{{1},{2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}


MATHEMATICA

Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], SubsetQ[#, Union@@@Select[Tuples[#, 2], Intersection@@#!={}&]]&]], {n, 0, 3}]


CROSSREFS

The case without singletons is A072446.
The unlabeled case is A326867.
The connected case is A326868.
Binomial transform of A326870 (the covering case).
The BIInumbers of these setsystems are A326872.
Cf. A072444, A072447, A102896, A306445.
Sequence in context: A137704 A001417 A156926 * A001697 A006069 A270485
Adjacent sequences: A326863 A326864 A326865 * A326867 A326868 A326869


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Jul 29 2019


STATUS

approved



