Lattice paths and submonoids of Z2

We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set X⊆ Z², there are two naturally associated monoids: F_X, the monoid of all X-walks/paths; and A_X, the monoid of all endpoints of X-walks starting from the origin O. For each A∈ A_X, write π_X(A) for the number of X-walks from O to A. Calculating the numbers π_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 π_X(A) , geometrical properties of the step set X, algebraic properties of the monoid 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.