Geodesic
On some generalized $q$-Eulerian polynomials
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013).

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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. 
@article{DMTCS_2013_special_264_a93,
     author = {Lin, Zhicong},
     title = {On some generalized $q${-Eulerian} polynomials},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)},
     year = {2013},
     doi = {10.46298/dmtcs.12822},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.12822/}
}
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Lin, Zhicong. On some generalized $q$-Eulerian polynomials. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013). doi : 10.46298/dmtcs.12822. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.12822/

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