Lattice Paths and Submonoids of $$\mathbb Z^2$$

Abstract

We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set \(X\subseteq \mathbb Z^2\), there are two naturally associated monoids: \(\mathscr {F}_X\), the monoid of all X-walks/paths; and \(\mathscr {A}_X\), the monoid of all endpoints of X-walks starting from the origin O. For each \({A\in \mathscr {A}_X}\), write \(\pi _X(A)\) for the number of X-walks from O to A. Calculating the numbers \(\pi _X(A)\) is a classical problem, leading to Fibonacci, Catalan, Motzkin, Delannoy and Schröder numbers, among many other well-studied sequences and arrays. Our main results give relationships between finiteness properties of the numbers \(\pi _X(A)\), geometrical properties of the step set X, algebraic properties of the monoid \(\mathscr {A}_X\), and combinatorial properties of a certain bi-labelled digraph naturally associated to X. There is an intriguing divergence between the cases of finite and infinite step sets, and some constructions rely on highly non-trivial properties of real numbers. We also consider the case of walks constrained to stay within a given region of the plane. Several examples are considered throughout to highlight the sometimes-subtle nature of the theoretical results.