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A326870
Number of connectedness systems covering n vertices.
13
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
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.
Sequence in context: A009485 A188455 A015056 * A327903 A214443 A015973
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 29 2019
EXTENSIONS
a(6) corrected by Christian Sievers, Oct 28 2023
STATUS
approved