Geodesic
On some generalized \(q\)-Eulerian polynomials
Zhicong Lin1
1Institut Camille Jordan, Université Lyon 1
The electronic journal of combinatorics, Tome 20 (2013) no. 1
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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.
DOI : 10.37236/2927
Classification : 05A05, 05A19
Mots-clés : Eulerian numbers, symmetrical Eulerian identities, hook factorization, descents, admissible inversions, permutation statistics

Zhicong Lin  1

1 Institut Camille Jordan, Université Lyon 1 
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     title = {On some generalized {\(q\)-Eulerian} polynomials},
     journal = {The electronic journal of combinatorics},
     year = {2013},
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     doi = {10.37236/2927},
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     url = {http://geodesic.mathdoc.fr/articles/10.37236/2927/}
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Zhicong Lin. On some generalized \(q\)-Eulerian polynomials. The electronic journal of combinatorics, Tome 20 (2013) no. 1. doi: 10.37236/2927

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