Interlaced rectangular parking functions

Abstract

The aim of this work is to extend to a general $S_m\times S_n$-module context the Grossman-Bizley paradigm that allows the enumeration of Dyck paths in a $m\times n$-rectangle. We obtain an explicit formula for the the "bi-Frobenius" characteristic of what we call {\em interlaced} rectangular parking functions in an $m\times n$-rectangle. These are obtained by labelling the $n$ vertical steps of an $m\times n$-Dyck path by the numbers from $1$ to $n$, together with an independent labelling of its horizontal steps by integers from $1$ to $m$. Our formula specializes to give the Frobenius characteristic of the $S_n$-module of $m\times n$-parking functions in the general situation. Hence, it subsumes the result of Armstrong-Loehr-Warrington which furnishes such a formula for the special case when $m$ and $n$ are coprime integers.