Towards a human proof of Gessel's conjecture

TL;DR

This paper explores walks in the first quadrant with specific steps, providing explicit formulas and connections to Dyck paths and cyclic polygons, aiming towards a human proof of Gessel's conjecture.

Contribution

It introduces a novel interpretation of quadrant walks as generalized Dyck words and establishes explicit formulas linking these to cyclic polygon area polynomials.

Findings

Explicit formula for certain quadrant walks

Connection between walks and Dyck paths

Identified a mysterious link to cyclic polygon area polynomials

Abstract

We interpret walks in the first quadrant with steps {(1,1),(1,0),(-1,0), (-1,-1)} as a generalization of Dyck words with two sets of letters. Using this language, we give a formal expression for the number of walks in the steps above beginning and ending at the origin. We give an explicit formula for a restricted class of such words using a correspondance between such words and Dyck paths. This explicit formula is exactly the same as that for the degree of the polynomial satisfied by the square of the area of cyclic n-gons conjectured by Dave Robbins although the connection is a mystery. Finally we remark on another combinatorial problem in which the same formula appears and argue for the existence of a bijection.