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Combinatorial relations among restricted and half Eulerian polynomials of types $A$, $B$, and $D$

Zhong-Xue Zhang

Abstract

In this paper, we study relations among several types of Eulerian polynomials from a combinatorial viewpoint. We establish an identity between the restricted Eulerian polynomials of types $A$ and $B$. As an application, we present a bijective proof of a new identity involving the Eulerian polynomials of type $A$ and type $B$, solving a recent open problem proposed by Zhang. Additionally, we derive an identity between the half Eulerian polynomials of type $B$ and type $D$. Using this identity, we further obtain another relation about the Eulerian polynomials of type $A$ and type $B$, as well as a recursive formula connecting the restricted Eulerian polynomials of type $D$ and Eulerian polynomials of types $A$ and $B$.

Combinatorial relations among restricted and half Eulerian polynomials of types $A$, $B$, and $D$

Abstract

In this paper, we study relations among several types of Eulerian polynomials from a combinatorial viewpoint. We establish an identity between the restricted Eulerian polynomials of types and . As an application, we present a bijective proof of a new identity involving the Eulerian polynomials of type and type , solving a recent open problem proposed by Zhang. Additionally, we derive an identity between the half Eulerian polynomials of type and type . Using this identity, we further obtain another relation about the Eulerian polynomials of type and type , as well as a recursive formula connecting the restricted Eulerian polynomials of type and Eulerian polynomials of types and .
Paper Structure (3 sections, 14 theorems, 69 equations)

This paper contains 3 sections, 14 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Bijective proofs of Theorem \ref{['thm-Zhang']} and Theorem \ref{['thm-AB']}
  3. Proof of Theorem \ref{['thm-BDhalf']}

Key Result

Theorem 1.1

For positive integer $n$, we have

Theorems & Definitions (29)

  • Theorem 1.1: Zhang25
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Example 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['thm-AB']}
  • Lemma 2.4
  • ...and 19 more