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

TL;DR

This paper explores combinatorial relations among Eulerian polynomials of types A, B, and D, establishing new identities and bijective proofs that connect these polynomial families and solve an open problem.

Contribution

It introduces new identities and bijective proofs relating restricted and half Eulerian polynomials of types A, B, and D, advancing understanding of their combinatorial relations.

Findings

Established an identity between restricted Eulerian polynomials of types A and B.

Provided a bijective proof for a new identity involving types A and B.

Derived a recursive formula connecting types D, A, and B Eulerian polynomials.

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 $B$.