Classifying lattice walks restricted to the quarter plane

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. Introduction The interest in an enumerative approach to lattice walks under various different types of restrictions has risen recently, (see [4, 6, 7, 19]) and there is increased need for global approaches and results. A few such studies have been performed. For example, in the case of the one dimensional lattice walks, Flajolet and Banderier [1] examine the nature of their generating functions, and provide general results of asymptotic analysis. For two dimensional walks in the quarter plane, we find a collection of case analyses [6, 7, 12, 17], and the goal here is to try to determine more general characterizations of these walks, based on the nature of their generating functions. Essentially we are interested to know which walks have holonomic generating functions, that is, when does the generating function satisfy systems of independent linear differential equations with polynomial coefficients. The answer has important repercussions for sequence generation, asymptotics, amongst other enumerative questions. Unfortunately we do not completely succeed in giving a characterization of all walks, but we do uncover some interesting patterns, and present variety of applications of the kernel method. We derive enumerative information to support a new conjecture on the combinatorial conditions required for a class of walks to possess holonomic (or, D-finte) generating functions. 1. Walks and their generating functions 1.1. Next nearest neighbour walks. The walks of interest here are known as next nearest neighbour walks. Precisely, they use movements on the integer lattice where each step is from some fixed set Y ⊆ {±1, 0}2 \ {(0, 0)}, which we also specify by the compass directions {N,NE, ...,W,NW}. Such a set Y is called a step set. Here we shall consider nearest neighbour walks exclusively, unless explicitly mentioned otherwise. A walk in the quarter plane is a sequence of steps w in Y∗, 2000 Mathematics Subject Classification. Primary 82B41; Secondary 05A15.