Increasing Binary Trees and the (α,β)-Eulerian Polynomials

2024/04/16 by William Y. C. Chen, Chen, William Y. C., Amy M. Fu +1
Computer Science · Mathematics · #05A15 #05A19 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2404.10331

openalex publication_date 2024/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In light of the grammar given by Ji for the (α,β)-Eulerian polynomials introduced by Carlitz and Scoville, we provide a labeling scheme for increasing binary trees. In this setting, we obtain a combinatorial interpretation of the γ-coefficients of the α-Eulerian polynomials in terms of forests of planted 0-1-2-plane trees, which specializes to a combinatorial interpretation of the γ-coefficients of the derangement polynomials in the same vein. By means of a decomposition of an increasing binary tree into a forest, we find combinatorial interpretations of the sums involving two identities of Ji, one of which can be viewed as (α,β)-extensions of the formulas of Petersen and Stembridge.

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