A solution to the tennis ball problem

TL;DR

This paper provides an explicit solution to the tennis ball problem by deriving generating functions using Tutte polynomials, and extends the method to broader classes of lattice path problems.

Contribution

It introduces a novel approach using Tutte polynomials to solve lattice path counting problems and generalizes the solution to wider problem classes.

Findings

Explicit generating functions for the tennis ball problem.

Method applicable to a broad class of lattice path problems.

Extension of the solution to generalized boundary conditions.

Abstract

We present a complete solution to the so-called tennis ball problem, which is equivalent to counting lattice paths in the plane that use North and East steps and lie between certain boundaries. The solution takes the form of explicit expressions for the corresponding generating functions. Our method is based on the properties of Tutte polynomials of matroids associated to lattice paths. We also show how the same method provides a solution to a wide generalization of the problem.