Explicit expression for the generating function counting Gessel's walks

TL;DR

This paper derives an explicit formula for the generating function that counts Gessel's walks, a specific class of lattice paths in the positive quadrant with four types of steps, starting at the origin and ending at a given point.

Contribution

The paper provides the first explicit expression for the generating function of Gessel's walks, advancing combinatorial enumeration methods for lattice path problems.

Findings

Explicit generating function derived for Gessel's walks

Enables precise counting of walks ending at specific points

Facilitates further combinatorial and probabilistic analysis

Abstract

Gessel's walks are the planar walks that move within the positive quadrant $\mathbb{Z}_{+}^{2}$ by unit steps in any of the following directions: West, North-East, East and South-West. In this paper, we find an explicit expression for the trivariate generating function counting the Gessel's walks with $k\geq 0$ steps, which start at $(0,0)$ and end at a given point $(i,j) \in \mathbb{Z}^2_+$.