A note on the $\gamma$-coefficients of the "tree Eulerian polynomial"

TL;DR

This paper explores the properties of a polynomial counting rooted trees by descending edges, extending the Eulerian polynomial, and provides combinatorial interpretations of its gamma-coefficients.

Contribution

It extends the classical Eulerian polynomial to rooted trees, proves its factorization over integers, and derives combinatorial interpretations of its gamma-coefficients.

Findings

The polynomial factors completely over the integers.

Gamma-coefficients are positive and have combinatorial interpretations.

Connections to binary trees and Stirling permutations are established.

Abstract

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. B. Drake proved that this polynomial factors completely over the integers. From his product formula it can be concluded that this polynomial has positive coefficients in the $\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these positive coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of the author and Wachs related to the poset of weighted partitions and the free multibracketed Lie algebra.