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

William Y. C. Chen, Amy M. Fu

TL;DR

This work develops a labeling framework on increasing binary trees—the $(a,b,\alpha,\beta)$-labeling—that encodes permutation statistics and yields a combinatorial interpretation of the $\gamma$-coefficients for the $(\alpha,\beta)$-Eulerian polynomials via forests of planted $0$-$1$-$2$-plane trees. By decomposing trees into supporting forests and exploiting weight-assigning rules, the authors obtain gamma-expansions for the $\alpha$-Eulerian polynomials and derive corresponding results for derangement polynomials, including a $q$-analogue linked to cycle structure. The paper also provides two labeling schemes for interior peaks that translate Ji's identities into equivalences between sums over planted forest structures, delivering a direct combinatorial proof and a cancellation argument for a derangement-related identity. Overall, the approach unifies grammar-based and tree-based combinatorics to illuminate $\gamma$-positivity and structural decompositions across Eulerian and derangement polynomials. These results enhance understanding of how planted forest decompositions encode refined permutation statistics and extend combinatorial interpretations of classical polynomial families via labeled increasing binary trees.

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.

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

TL;DR

This work develops a labeling framework on increasing binary trees—the -labeling—that encodes permutation statistics and yields a combinatorial interpretation of the -coefficients for the -Eulerian polynomials via forests of planted ---plane trees. By decomposing trees into supporting forests and exploiting weight-assigning rules, the authors obtain gamma-expansions for the -Eulerian polynomials and derive corresponding results for derangement polynomials, including a -analogue linked to cycle structure. The paper also provides two labeling schemes for interior peaks that translate Ji's identities into equivalences between sums over planted forest structures, delivering a direct combinatorial proof and a cancellation argument for a derangement-related identity. Overall, the approach unifies grammar-based and tree-based combinatorics to illuminate -positivity and structural decompositions across Eulerian and derangement polynomials. These results enhance understanding of how planted forest decompositions encode refined permutation statistics and extend combinatorial interpretations of classical polynomial families via labeled increasing binary trees.

Abstract

In light of the grammar given by Ji for the -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 -coefficients of the -Eulerian polynomials in terms of forests of planted 0-1-2-plane trees, which specializes to a combinatorial interpretation of the -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 -extensions of the formulas of Petersen and Stembridge.
Paper Structure (7 sections, 10 theorems, 20 equations, 6 figures)

This paper contains 7 sections, 10 theorems, 20 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The $(\alpha,\beta)$-Eulerian polynomials
  3. The $(a,b,\alpha,\beta)$-labeling
  4. A decomposition
  5. The $\alpha$-Eulerian polynomials
  6. The derangement polynomials
  7. A labeling scheme for interior peaks

Key Result

Theorem 2.1

Let $D$ denote the formal derivative of the grammar $G$. For $n\geq 0$, we have

Figures (6)

  • Figure 1: An example for the $(a,b,\alpha,\beta)$-labeling.
  • Figure 2: A supporting forest on $[2,n]$ with inherited labels.
  • Figure 3: A supporting forest with updated labels.
  • Figure 4: A forest of planted 0-1-2-plane trees.
  • Figure 5: The first modified $(a,b,\alpha,\beta)$-labeling.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 2.1: Ji
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 3.1: Ji
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4: Ji