%I #11 Oct 28 2023 12:49:04

%S 1,1,5,77,6377,8097721,1196051135917

%N Number of connectedness systems covering n vertices.

%C 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 covering if every vertex belongs to some edge.

%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.

%e The a(2) = 5 connectedness systems:

%e {{1,2}}

%e {{1},{2}}

%e {{1},{1,2}}

%e {{2},{1,2}}

%e {{1},{2},{1,2}}

%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&SubsetQ[#,Union@@@Select[Tuples[#,2],Intersection@@#!={}&]]&]],{n,0,4}]

%Y Inverse binomial transform of A326866 (the non-covering case).

%Y Exponential transform of A326868 (the connected case).

%Y The unlabeled case is A326871.

%Y The BII-numbers of these set-systems are A326872.

%Y The case without singletons is A326877.

%Y Cf. A072445, A072446, A072447, A102896, A323818, A326867, A326869.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Jul 29 2019

%E a(6) corrected by _Christian Sievers_, Oct 28 2023