OFFSET
1,3
COMMENTS
Previous name was: a(1) = 1; for n > 1, a(n) = number of families of subsets of {1, ..., n} that contain both the universe and the empty set, are closed under union of nondisjoint sets, and contain no singletons.
A connectedness system is (as below) a set of (finite) nonempty sets that is closed under union of nondisjoint sets.
The old definition was: "Number of subsets S of the power set P{1,2,...,n} such that: {1}, {2},..., {n} are all elements of S; {1,2,...n} is an element of S; if X and Y are elements of S and X and Y have a nonempty intersection, then the union of X and Y is an element of S."
Comments on the old definition from Gus Wiseman, Aug 01 2019: (Start)
If this sequence were defined similarly to A326877, we would have a(1) = 0.
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. It is connected if it is empty or contains an edge with all the vertices. a(n) is the number of connected connectedness systems on n vertices without singletons. For example, the a(3) = 8 connected connectedness systems without singletons are:
{{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)
Conjecture concerning the original definition: a(n) is also the number of families of subsets of {1, ..., n} that contain both the universe and the empty set, are closed under intersection and contain no sets of cardinality n-1. - Tian Vlasic, Nov 04 2022. [This was false, as pointed out by Christian Sievers, Oct 20 2023. It is easy to see that for n>1, a(n) is also the number of families of subsets of {1, ..., n} that contain both the universe and the empty set, are closed under union of nondisjoint sets, and contain no singletons; whereas by duality, the sequence suggested in the conjecture is also the number of those families that are also closed under arbitrary union. For details see the Sievers link. - N. J. A. Sloane, Oct 21 2023]
LINKS
Christian Sievers, Comments on connectedness systems: the conjecture about A072447
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 > 1) = A326868(n)/2^n. - Gus Wiseman, Aug 01 2019
EXAMPLE
a(3) = 8 because of the 8 sets: {{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}]], (n==0||MemberQ[#, Range[n]])&&SubsetQ[#, Union@@@Select[Tuples[#, 2], Intersection@@#!={}&]]&]], {n, 0, 4}] (* returns a(1) = 0 similar to A326877. - Gus Wiseman, Aug 01 2019 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Wim van Dam (vandam(AT)cs.berkeley.edu), Jun 18 2002
EXTENSIONS
Edited by N. J. A. Sloane, Oct 21 2023 (a(6) corrected by Christian Sievers, Oct 20 2023)
Edited by Christian Sievers, Oct 26 2023
STATUS
approved