%I #12 Oct 27 2023 23:39:41
%S 1,2,8,96,6720,8130432,1196099819520
%N Number of connectedness systems on 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 two overlapping edges.
%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) = 2^n * A072446(n).
%e The a(0) = 1 through a(2) = 8 connectedness systems:
%e {} {} {}
%e {{1}} {{1}}
%e {{2}}
%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}]],SubsetQ[#,Union@@@Select[Tuples[#,2],Intersection@@#!={}&]]&]],{n,0,3}]
%Y The case without singletons is A072446.
%Y The unlabeled case is A326867.
%Y The connected case is A326868.
%Y Binomial transform of A326870 (the covering case).
%Y The BII-numbers of these set-systems are A326872.
%Y Cf. A072444, A072447, A102896, A306445.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Jul 29 2019
%E a(6) corrected by _Christian Sievers_, Oct 26 2023