A probabilistic approach to enumeration of Gessel walks

TL;DR

This paper introduces a probabilistic model for Gessel walks, deriving recurrence relations and polynomial formulas that resolve two conjectures posed by Petkovšek and Wilf.

Contribution

A novel probabilistic framework for Gessel walks enabling the derivation of recurrence relations and polynomial formulas, solving two longstanding conjectures.

Findings

Derived linear recurrence relations with binomial coefficients for Gessel walk counts.

Established polynomial formulas with integer coefficients for specific Gessel walk enumerations.

Solved two conjectures of Petkovšek and Wilf regarding Gessel walks.

Abstract

We consider Gessel walks in the plane starting at the origin $(0, 0)$ remaining in the first quadrant $i, j \geq 0$ and made of West, North-East, East and South-West steps. Let $F(m; n_1, n_2)$ denote the number of these walks with exact $m$ steps ending at the point $(n_1, n_2)$, Petkov\v{s}ek and Wilf posed several analogous conjectures similar to the famous Gessel's conjecture. We establish a probabilistic model of Gessel walks which is concerned with the problem of vicious walkers. This model helps us to obtain the linear homogeneous recurrence relations with binomial coefficients for both $F(n+k+r;n+k-r,n)$ and $F(n+2k; n, 0)$. Precisely, $\frac{n! k! (n+k+1)!}{(2n+2)!} F(2n+2k;0,n)$ is a polynomial with all integer coefficients which leading term is $2^{3k-2} n^{2k-2}$, and $\frac{k! (k+1)!}{n+1} F(n+2k;n,0)$ is a polynomial with all integer coefficients which leading term is $n^{2k-1}$. Hence two conjectures of Petkov\v{s}ek and Wilf are solved.