A327228 Number of set-systems with n vertices and at least one endpoint/leaf.

0, 1, 6, 65, 3297, 2537672, 412184904221, 4132070624893905681577, 174224571863520492218909428465944685216436, 133392486801388257127953774730008469745829658368044283629394202488602260177922751

Offset: 0

Author

Gus Wiseman, Sep 01 2019

Keywords

• nonn

Comments

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 set-systems with minimum covered vertex-degree 1.

Examples

			The a(2) = 6 set-systems:
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
		

Links

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

Crossrefs

The covering version is A327229.

The specialization to simple graphs is A245797.

BII-numbers of these set-systems are A327105.

Cf. A058891, A059167, A327098, A327103, A327104, A327107, A327197, A327227, A327230, A330059.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,4}]

Formula

Binomial transform of A327229.

a(n) = A058891(n+1) - A330059(n). - Andrew Howroyd, Jan 21 2023

Extensions

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