A Note on the γ-Coefficients of the Tree Eulerian Polynomial

We consider the generating polynomial of the number of rooted trees on the set {1,2,...,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. A classical product formula shows that this polynomial factors completely over the integers. From this product formula it can be concluded that this polynomial has positive coefficients in the γ-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these 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 Liu, Dotsenko-Khoroshkin, Bershtein-Dotsenko-Khoroshkin, González D\u27León-Wachs and González D\u27León related to the free multibracketed Lie algebra and the poset of weighted partitions

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

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.Comment: 13 pages, 6 figures, Interpretations derived from results in arXiv:1309.5527 and arXiv:1408.541

On two unimodal descent polynomials

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $\gamma$-coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted.Comment: 16 pages, 4 figure