On some generalized q-Eulerian polynomials

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.