On some generalized $q$-Eulerian polynomials

TL;DR

This paper introduces new bijective proofs and recurrence relations for generalized $(q,r)$-Eulerian polynomials, expanding understanding of their combinatorial properties and symmetries.

Contribution

It provides bijective proofs for $q$-Eulerian identities, a new recurrence formula, and studies a $q$-analogue of restricted descent polynomials.

Findings

Established a generalized symmetrical identity for restricted $q$-Eulerian polynomials.

Derived a new recurrence relation for $(q,r)$-Eulerian polynomials.

Provided combinatorial proofs for existing $q$-Eulerian identities.

Abstract

The $(q,r)$-Eulerian polynomials are the $(\maj-$$\exc,\fix,\exc)$ enumerative polynomials of permutations. Using Shareshian and Wachs' exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical $q$-Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han's three-variable statistic $(\inv-$$\lec,\pix,\lec)$. We also prove a new recurrence formula for the $(q,r)$-Eulerian polynomials and study a $q$-analogue of Chung and Graham's restricted descent polynomials. In particular, we obtain a generalized symmetrical identity for these restricted $q$-Eulerian polynomials with a combinatorial proof.