On some generalized $q$-Eulerian polynomials

Abstract : The $(q,r)$-Eulerian polynomials are the $(\mathrm{maj-exc, fix, exc})$ 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 $(\mathrm{inv-lec, pix, lec})$. We also prove a new recurrence formula for the $(q,r)$-Eulerian polynomials and study a $q$-analogue of Chung and Graham's restricted Eulerian polynomials. In particular, we obtain a symmetrical identity for these restricted $q$-Eulerian polynomials with a combinatorial proof.
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Zhicong Lin. On some generalized $q$-Eulerian polynomials. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.409-420. ⟨hal-01229749⟩

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