$q$-Counting Descent Pairs with Prescribed Tops and Bottoms

Abstract

Given sets $X$ and $Y$ of positive integers and a permutation $\sigma = \sigma_1 \sigma_2 \cdots \sigma_n \in S_n$, an $(X,Y)$-descent of $\sigma$ is a descent pair $\sigma_i > \sigma_{i+1}$ whose "top" $\sigma_i$ is in $X$ and whose "bottom" $\sigma_{i+1}$ is in $Y$. Recently Hall and Remmel proved two formulas for the number $P_{n,s}^{X,Y}$ of $\sigma \in S_n$ with $s$ $(X,Y)$-descents, which generalized Liese's results in [1]. We define a new statistic ${\rm stat}_{X,Y}(\sigma)$ on permutations $\sigma$ and define $P_{n,s}^{X,Y}(q)$ to be the sum of $q^{{\rm stat}_{X,Y}(\sigma)}$ over all $\sigma \in S_n$ with $s$ $(X,Y)$-descents. We then show that there are natural $q$-analogues of the Hall-Remmel formulas for $P_{n,s}^{X,Y}(q)$.