On some Generalized $q$-Eulerian Polynomials

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.