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A326868
Number of connected connectedness systems on n vertices.
8
1, 1, 4, 64, 6048, 8064000, 1196002238976
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 connected if it is empty or contains an edge with all the vertices.
LINKS
Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.
FORMULA
a(n > 1) = 2^n * A072447(n).
Logarithmic transform of A326870.
EXAMPLE
The a(3) = 64 connected connectedness systems:
{{123}} {{1}{123}}
{{12}{123}} {{2}{123}}
{{13}{123}} {{3}{123}}
{{23}{123}} {{1}{12}{123}}
{{12}{13}{123}} {{1}{13}{123}}
{{12}{23}{123}} {{1}{23}{123}}
{{13}{23}{123}} {{2}{12}{123}}
{{12}{13}{23}{123}} {{2}{13}{123}}
{{2}{23}{123}}
{{3}{12}{123}}
{{3}{13}{123}}
{{3}{23}{123}}
{{1}{12}{13}{123}}
{{1}{12}{23}{123}}
{{1}{13}{23}{123}}
{{2}{12}{13}{123}}
{{2}{12}{23}{123}}
{{2}{13}{23}{123}}
{{3}{12}{13}{123}}
{{3}{12}{23}{123}}
{{3}{13}{23}{123}}
{{1}{12}{13}{23}{123}}
{{2}{12}{13}{23}{123}}
{{3}{12}{13}{23}{123}}
.
{{1}{2}{123}} {{1}{2}{3}{123}}
{{1}{3}{123}} {{1}{2}{3}{12}{123}}
{{2}{3}{123}} {{1}{2}{3}{13}{123}}
{{1}{2}{12}{123}} {{1}{2}{3}{23}{123}}
{{1}{2}{13}{123}} {{1}{2}{3}{12}{13}{123}}
{{1}{2}{23}{123}} {{1}{2}{3}{12}{23}{123}}
{{1}{3}{12}{123}} {{1}{2}{3}{13}{23}{123}}
{{1}{3}{13}{123}} {{1}{2}{3}{12}{13}{23}{123}}
{{1}{3}{23}{123}}
{{2}{3}{12}{123}}
{{2}{3}{13}{123}}
{{2}{3}{23}{123}}
{{1}{2}{12}{13}{123}}
{{1}{2}{12}{23}{123}}
{{1}{2}{13}{23}{123}}
{{1}{3}{12}{13}{123}}
{{1}{3}{12}{23}{123}}
{{1}{3}{13}{23}{123}}
{{2}{3}{12}{13}{123}}
{{2}{3}{12}{23}{123}}
{{2}{3}{13}{23}{123}}
{{1}{2}{12}{13}{23}{123}}
{{1}{3}{12}{13}{23}{123}}
{{2}{3}{12}{13}{23}{123}}
MATHEMATICA
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], n==0||MemberQ[#, Range[n]]&&SubsetQ[#, Union@@@Select[Tuples[#, 2], Intersection@@#!={}&]]&]], {n, 0, 4}]
CROSSREFS
The case without singletons is A072447.
The not necessarily connected case is A326866.
The unlabeled case is A326869.
The BII-numbers of these set-systems are A326879.
Sequence in context: A348315 A053923 A359231 * A211214 A372813 A229867
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 29 2019
EXTENSIONS
a(6) corrected by Christian Sievers, Oct 28 2023
STATUS
approved