A326877 - OEIS

%I #11 Oct 28 2023 12:08:10

%S 1,0,1,8,381,252080,18687541309

%N Number of connectedness systems covering n vertices without singletons.

%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(3) = 8 covering connectedness systems without singletons:

%e   {{1,2,3}}

%e   {{1,2},{1,2,3}}

%e   {{1,3},{1,2,3}}

%e   {{2,3},{1,2,3}}

%e   {{1,2},{1,3},{1,2,3}}

%e   {{1,2},{2,3},{1,2,3}}

%e   {{1,3},{2,3},{1,2,3}}

%e   {{1,2},{1,3},{2,3},{1,2,3}}

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

%Y Inverse binomial transform of A072446 (the non-covering case).

%Y Exponential transform of A072447 if we assume A072447(1) = 0 (the connected case).

%Y The case with singletons is A326870.

%Y The BII-numbers of these set-systems are A326873.

%Y Cf. A072444, A072445, A102896, A323818, A326866, A326867, A326868, A326871, A326872.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Jul 30 2019

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