A072447 Number of connectedness systems on n vertices that contain all singletons and the set of all the vertices.

1, 1, 8, 378, 252000, 18687534984

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

Table of n, a(n) for n=1..6.

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

The unlabeled case is A072445.

The non-connected case is A072446.

The case with singletons is A326868.

The covering version is A326877.

Cf. A072444, A326866, A326869, A326870, A326873, A326879.

Sequence in context: A206690 A349114 A266920 * A326877 A332138 A225698

Adjacent sequences: A072444 A072445 A072446 * A072448 A072449 A072450

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