Bounce statistics for rational lattice paths

Abstract

Given two relatively prime positive integers $\alpha$ and $\beta$, we consider simple lattice paths (with unit East and unit North steps) from $(0,0)$ to $(\alpha k,\beta k)$, and enumerate them by their left and right bounces with respect to the line $y=\frac{\beta}{\alpha} x$. We give the corresponding multivariate generating functions for all such paths as well as for subclasses of paths that start and end with a prescribed step. For illustration purposes, we discuss the case $\beta=1$ and express some of our functions in terms of the Fuss-Catalan generating function $c_\alpha(x)$.