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.

Keywords

• eulerian numbers
• symmetrical eulerian identities
• hook factorization
• descents
• admissible inversions
• [info.info-dm]computer science [cs]/discrete mathematics [cs.dm]