Square lattice walks avoiding a quadrant

TL;DR

This paper studies square lattice walks avoiding specific non-convex quadrants, revealing their generating functions are algebraic or D-finite but transcendental, and provides explicit formulas for walk enumeration.

Contribution

It introduces new enumeration results for walks avoiding non-convex cones, showing their generating functions are algebraic or D-finite, and connects these to Gessel's walks.

Findings

Generating functions differ from simple D-finite series by algebraic parts.

Explicit sum formulas involving hypergeometric terms for walk counts.

Gessel's walks have algebraic or D-finite generating functions depending on the starting point.

Abstract

In the past decade, a lot of attention has been devoted to the enumera-tion of walks with prescribed steps confined to a convex cone. In two dimensions, this means counting walks in the first quadrant of the plane (possibly after a linear transformation). But what about walks in non-convex cones? We investigate the two most natural cases: first, square lattice walks avoiding the negative quadrant Q 1 = {(i, j) : i \textless{} 0 and j \textless{} 0}, and then, square lattice walks avoiding the West quadrant Q 2 = {(i, j) : i \textless{} j and i \textless{} --j}. In both cases, the generating function that counts walks starting from the origin is found to differ from a simple D-finite series by an algebraic one. We also obtain closed form expressions for the number of n-step walks ending at certain prescribed endpoints, as a sum of three hypergeometric terms. One of these terms already appears in the enumeration of square lattice walks confined to the cone {(i, j) : i +j $\ge$ 0 and j $\ge$ 0}, known as Gessel's walks. In fact, the enumeration of Gessel's walks follows, by the reflection principle, from the enumeration of walks starting from (--1, 0) and avoiding Q 1. Their generating function turns out to be purely algebraic (as the generating function of Gessel's walks). Another approach to Gessel's walks consists in counting walks that start from (--1, 1) and avoid the West quadrant Q 2. The associated generating function is D-finite but transcendental.