A remark on the enumeration of rooted labeled trees

Abstract

Two decades ago, Chauve, Dulucq and Guibert showed that the number of rooted trees on the vertex set $[n+1]$ in which exactly $k$ children of the root are lower-numbered than the root is $\left(\genfrac{}{}{0ex}{}{n}{k}\right)n^{n-k}$. Here I give a simpler proof of this result.

LaTex2e, 9 pages. Version 2 contains a Note Added with a quick and elegant proof due to Jiang Zeng. To be published in Discrete Mathematics

References in corpus (1)

• A recursive algorithm for trees and forests

Cited by in corpus (2)

• Total positivity of some polynomial matrices that enumerate labeled trees and forests, I. Forests of rooted labeled trees
• Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs