Bounce statistics for rational lattice paths

TL;DR

This paper analyzes lattice paths between specific points, counting bounce events relative to a line, and provides generating functions for these paths, including special cases related to Fuss-Catalan numbers.

Contribution

It introduces multivariate generating functions for bounce statistics of rational lattice paths and explores subclasses with fixed start and end steps.

Findings

Derived explicit generating functions for bounce statistics

Connected bounce counts to Fuss-Catalan generating functions

Analyzed special cases with fixed start and end steps

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 $c_\alpha(x)$.