On some generalized \(q\)-Eulerian polynomials (Q1953442)

Summary: The (\(q,r\))-Eulerian polynomials are the (\textsf{maj-exc}, \textsf{fix}, \textsf{exc}) enumerative polynomials of permutations. Using Shareshian and Wachs' exponential generating function of these Eulerian polynomials, \textit{F. Chung} and \textit{R. Graham} [J. Comb. 3, 299--316 (2012)] proved two symmetrical \(q\)-Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han's three-variable statistic (\textsf{inv-lec}, \textsf{pix}, \textsf{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.