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Number of connectedness systems on n vertices that contain all singletons and the set of all the vertices.
15

%I #44 Oct 28 2023 11:12:14

%S 1,1,8,378,252000,18687534984

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

%C 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.

%C A connectedness system is (as below) a set of (finite) nonempty sets that is closed under union of nondisjoint sets.

%C 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."

%C Comments on the old definition from _Gus Wiseman_, Aug 01 2019: (Start)

%C If this sequence were defined similarly to A326877, we would have a(1) = 0.

%C 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:

%C {{1,2,3}}

%C {{1,2},{1,2,3}}

%C {{1,3},{1,2,3}}

%C {{2,3},{1,2,3}}

%C {{1,2},{1,3},{1,2,3}}

%C {{1,2},{2,3},{1,2,3}}

%C {{1,3},{2,3},{1,2,3}}

%C {{1,2},{1,3},{2,3},{1,2,3}}

%C (End)

%C 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]

%H Christian Sievers, <a href="/A072447/a072447_1.txt">Comments on connectedness systems: the conjecture about A072447</a>

%H Wim van Dam, <a href="http://www.cs.berkeley.edu/~vandam/subpowersets/sequences.html">Sub Power Set Sequences</a>

%H Gus Wiseman, <a href="http://www.mathematica-journal.com/2017/12/every-clutter-is-a-tree-of-blobs/">Every Clutter Is a Tree of Blobs</a>, The Mathematica Journal, Vol. 19, 2017.

%F a(n > 1) = A326868(n)/2^n. - _Gus Wiseman_, Aug 01 2019

%e 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}}.

%t 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 *)

%Y The unlabeled case is A072445.

%Y The non-connected case is A072446.

%Y The case with singletons is A326868.

%Y The covering version is A326877.

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

%K nonn,more

%O 1,3

%A Wim van Dam (vandam(AT)cs.berkeley.edu), Jun 18 2002

%E Edited by _N. J. A. Sloane_, Oct 21 2023 (a(6) corrected by _Christian Sievers_, Oct 20 2023)

%E Edited by _Christian Sievers_, Oct 26 2023