The complete Generating Function for Gessel Walks is Algebraic

TL;DR

This paper proves that the generating function counting Gessel walks in the quarter plane, starting at the origin and with steps from a specific set, is algebraic, revealing a deep combinatorial structure.

Contribution

It establishes that the generating series for Gessel walks is algebraic, providing a complete generating function characterization.

Findings

The generating series G(t;x,y) is algebraic.

Gessel walks are counted by an algebraic generating function.

The result advances understanding of lattice path enumeration in constrained regions.

Abstract

Gessel walks are lattice walks in the quarter plane $\set N^2$ which start at the origin $(0,0)\in\set N^2$ and consist only of steps chosen from the set $\{\leftarrow,\swarrow,\nearrow,\to\}$. We prove that if $g(n;i,j)$ denotes the number of Gessel walks of length $n$ which end at the point $(i,j)\in\set N^2$, then the trivariate generating series $G(t;x,y)=\sum_{n,i,j\geq 0} g(n;i,j)x^i y^j t^n$ is an algebraic function.