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.
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
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 29 2019
EXTENSIONS
a(6) corrected by Christian Sievers, Oct 28 2023
STATUS
approved