A326866 Number of connectedness systems on n vertices.

1, 2, 8, 96, 6720, 8130432, 1196099819520

Offset: 0

Author

Gus Wiseman, Jul 29 2019

Keywords

• nonn
• more

Comments

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.

Examples

			The a(0) = 1 through a(2) = 8 connectedness systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}
		

Links

• Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Crossrefs

The case without singletons is A072446.

The unlabeled case is A326867.

The connected case is A326868.

Binomial transform of A326870 (the covering case).

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

Cf. A072444, A072447, A102896, A306445.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Union@@@Select[Tuples[#,2],Intersection@@#!={}&]]&]],{n,0,3}]

Formula

a(n) = 2^n * A072446(n).

Extensions

a(6) corrected by Christian Sievers, Oct 26 2023