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 the strong $q$-log-convexity of polynomials sequences, whose generating functions can be given by the continued fraction. As consequences, we get the strong $q$-log-convexity the Eulerian polynomials of type $A_n,B_n$, their $q$-analogous and the generalized Eulerian polynomials associated to the arithmetic progression $\{a,a+d,a+2d,a+3d,\ldots\}$ in a unified manner.
对于任意秩为$n$的Coxeter系统$(W,S)$，我们引入一个维数为$2n-1$的抽象布尔复形（单纯偏序集），它包含Coxeter复形作为相对子复形。面由三元组$(I,w,J)$索引，其中$I$和$J$是简单生成元集合$S$的子集，$w$是抛物线双陪集$W_I w W_J$的最小长度代表。对于群$W$中的每个元素，恰好有一个最大面。复形是薄的可壳的，这意味着复形是有限Coxeter群的球面。在这种情况下，$h$ -多项式的自然细化由“双侧”$W$ -欧拉多项式给出，即，$W$中左和右下降联合分布的生成函数。
Eulerian quasisymmetric functions and poset topology
We introduce a family of quasisymmetric functions called { \em Eulerian quasisymmetric functions}, which have the property of specializing to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializ…
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 fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surpris…
