A327229 Number of set-systems covering n vertices with at least one endpoint/leaf.

0, 1, 4, 50, 3069, 2521782, 412169726428, 4132070622008664529903, 174224571863520492185852863478334475199686, 133392486801388257127953774730008469744261637221272599199572772174870315402893538

Offset: 0

Author

Gus Wiseman, Sep 01 2019

Keywords

• nonn

Comments

Covering means there are no isolated vertices.

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.

Also covering set-systems with minimum vertex-degree 1.

Examples

			The a(2) = 4 set-systems:
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..12

Crossrefs

The non-covering version is A327228.

The specialization to simple graphs is A327227.

The unlabeled version is A327230.

BII-numbers of these set-systems are A327105.

Cf. A003465, A245797, A327079, A327098, A327103, A327107, A327197.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,3}]

Formula

Inverse binomial transform of A327228.

Extensions

Terms a(5) and beyond from Andrew Howroyd, Jan 21 2023