OFFSET
0,3
COMMENTS
From Gus Wiseman, Jul 31 2019: (Start)
If we define a connectedness system to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges, then a(n) is the number of connectedness systems on n vertices without singleton edges. The BII-numbers of these set-systems are given by A326873. The a(3) = 12 connectedness systems without singletons are:
{}
{{1,2}}
{{1,3}}
{{2,3}}
{{1,2,3}}
{{1,2},{1,2,3}}
{{1,3},{1,2,3}}
{{2,3},{1,2,3}}
{{1,2},{1,3},{1,2,3}}
{{1,2},{2,3},{1,2,3}}
{{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}
(End)
LINKS
Wim van Dam, Sub Power Set Sequences
Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.
FORMULA
a(n) = A326866(n)/2^n. - Gus Wiseman, Jul 31 2019
EXAMPLE
a(3)=12 because of the 12 sets:
{{1}, {2}, {3}};
{{1}, {2}, {3}, {1, 2}};
{{1}, {2}, {3}, {1, 3}};
{{1}, {2}, {3}, {2, 3}};
{{1}, {2}, {3}, {1, 2, 3}};
{{1}, {2}, {3}, {1, 2}, {1, 2, 3}};
{{1}, {2}, {3}, {1, 3}, {1, 2, 3}};
{{1}, {2}, {3}, {2, 3}, {1, 2, 3}};
{{1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}};
{{1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}};
{{1}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}};
{{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.
MATHEMATICA
Table[Length[Select[Subsets[Subsets[Range[n], {2, n}]], SubsetQ[#, Union@@@Select[Tuples[#, 2], Intersection@@#!={}&]]&]], {n, 0, 3}] (* Gus Wiseman, Jul 31 2019 *)
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Wim van Dam (vandam(AT)cs.berkeley.edu), Jun 18 2002
EXTENSIONS
a(6) corrected and definition reformulated by Christian Sievers, Oct 26 2023
a(0)=1 prepended by Sean A. Irvine, Oct 02 2024
STATUS
approved