Classifying lattice walks restricted to the quarter plane

TL;DR

This paper classifies the generating functions of quarter-plane lattice walks with three-step sets, identifying algebraic, transcendental holonomic, and non-holonomic classes, and provides enumerative data supporting conjectures on walk classifications.

Contribution

It offers a complete classification of generating functions for these walks, introducing new algebraic and non-holonomic classes and supporting combinatorial conjectures.

Findings

Identified a new algebraic class related to Kreweras' walks

Discovered two new non-holonomic classes

Provided enumerative data on various walk classes

Abstract

This work considers lattice walks restricted to the quarter plane, with steps taken from a set of cardinality three. We present a complete classification of the generating functions of these walks with respect to the classes algebraic, transcendental holonomic and non-holonomic. The principal results are a new algebraic class related to Kreweras' walks; two new non-holonomic classes; and enumerative data on some other classes. These results provide strong evidence for conjectures which use combinatorial criteria to classify the generating functions all nearest neighbour walks in the quarter plane.