Related papers: On some generalized $q$-Eulerian polynomials

In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or…

We find a combinatorial interpretation of Shareshian and Wachs' $q$-binomial-Eulerian polynomials, which leads to an alternative proof of their $q$-$\gamma$-positivity using group actions. Motivated by the sign-balance identity of…

In 2008 Br\"and\'en proved a $(p,q)$-analogue of the $\gamma$-expansion formula for Eulerian polynomials and conjectured the divisibility of the $\gamma$-coefficient $\gamma_{n,k}(p,q)$ by $(p+q)^k$. As a follow-up, in 2012 Shin and Zeng…

This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the…

An identity of Chung, Graham and Knuth involving binomial coefficients and Eulerian numbers motivates our study of a class of polynomials that we call binomial-Eulerian polynomials. These polynomials share several properties with the…

We find a $q$-analog of the following symmetrical identity involving binomial coefficients $\binom{n}{m}$ and Eulerian numbers $A_{n,m}$, due to Chung, Graham and Knuth [{\it J. Comb.}, {\bf 1} (2010), 29--38]: {equation*} \sum_{k\geq…

We study two generalizations of the gamma-expansion of Eulerian polynomials from the viewpoint of the decompositions of statistics. We first present an expansion formula of the trivariate Eulerian polynomials, which are the enumerators for…

The classical Eulerian polynomials can be expanded in the basis $t^{k-1}(1+t)^{n+1-2k}$ ($1\leq k\leq\lfloor (n+1)/2\rfloor$) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian…

A symmetry of $(t,q)$-Eulerian numbers of type $B$ is combinatorially proved by defining an involution preserving many important statistics on the set of permutation tableaux of type $B$. This involution also proves a symmetry of the…

On the set of permutations of a finite set, we construct a bijection which maps the 3-vector of statistics $(maj-exc,des,exc)$ to a 3-vector $(maj\_2,\widetilde{des\_2},inv\_2)$ associated with the $q$-Eulerian polynomials introduced by…

The object of this paper is to give a systematic treatment of excedance-type polynomials. We first give a sufficient condition for a sequence of polynomials to have alternatingly increasing property, and then we present a systematic study…

We prove several identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials, extending results of Stembridge, Petersen, and Br\"and\'en. Additionally, we find $q$-exponential…

The Stirling permutations introduced by Gessel-Stanley have recently received considerable attention. Motivated by Ji's work on $(\alpha,\beta)$-Eulerian polynomials (Sci China Math., 2025) and Yan-Yang-Lin's work on $1/k$-Eulerian…

We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of…

Remixed Eulerian numbers are a polynomial $q$-deformation of Postnikov's mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such…

In the present article, we introduce a $(p,q)$-analogue of the poly-Euler polynomials and numbers by using the $(p,q)$-polylogarithm function. These new sequences are generalizations of the poly-Euler numbers and polynomials. We give…

In this paper we prove the strong $q$-log-convexity of the Eulerian polynomials of Coxeter groups using their exponential generating functions. Our proof is based on the theory of exponential Riordan arraya and a criterion for determining…

We prove two conjectures of Shareshian and Wachs about Eulerian quasisymmetric functions and polynomials. The first states that the cycle type Eulerian quasisymmetric function $Q_{\lambda,j}$ is Schur-positive, and moreover that the…

We define a generalization of the Eulerian polynomials and the Eulerian numbers by considering a descent statistic on segmented permutations coming from the study of 2-species exclusion processes and a change of basis in a Hopf algebra. We…

By the symmetric properties of Drichlet's type multiple q-l-function, we establish various identities concerning the generalized higher-order q-Euler polynomials. Furthermore, we give some interesting relationship between the power sums and…