Descents of Permutations in a Ferrers Board

Abstract

The classical Eulerian polynomials are defined by setting $$A_n(t)= \sum_{\sigma \in \mathfrak{S}_n} t^{1+\mathrm{des}(\sigma)}= \sum_{k=1}^n A_{n,k} t^k$$ where  $A_{n,k}$ is the number of permutations of length $n$ with $k-1$ descents. Let $A_n(t, q) = \sum_{\pi \in \mathfrak{S}_n} t^{1+{\rm des}(\pi)}q^{{\rm inv}(\pi)}$ be the $\mathrm{inv}$ $q$-analogue of the classical Eulerian polynomials whose generating function is well known: $$\begin{eqnarray}\sum_{n \geq 0} \frac{u^n A_n(t, q)}{[n]_q!} = \frac{1}{\displaystyle 1-t\sum_{k \geq 1} \frac{(1-t)^ku^k}{[k]_q!}}.\qquad\qquad(*)\label{perm_gf abs}\end{eqnarray}$$ In this paper we consider permutations  restricted in a Ferrers board and study their descent polynomials. For a general Ferrers board $F$, we derive a formula in the form of permanent for the restricted $q$-Eulerian polynomial $$A_F(t,q) := \sum_{\sigma \in F} t^{1+{\rm des}(\sigma)} q^{{\rm inv}(\sigma)}.$$ If the Ferrers board has the special shape of an $n\times n$ square  with a triangular board of size $s$ removed, we prove that $A_F(t,q)$ is  a sum of $s+1$ terms, each satisfying an equation that is similar to (*).   Then we apply our results to permutations with bounded drop (or excedance) size, for which the descent polynomial was first studied by Chung et al. (European J. Combin., 31(7) (2010):1853-1867). Our method presents an alternative approach.