2018/11/14 by East, James, Ham, Nicholas
#Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1811.05735
We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set X⊆\mathbb Z2, there are two naturally associated monoids: \mathscr FX, the monoid of all X-walks/paths; and \mathscr AX, the monoid of all endpoints of X-walks starting from the origin O. For each A∈\mathscr AX, write πX(A) for the number of X-walks from O to A. Calculating the numbers πX(A) is a classical problem, leading to Fibonacci, Catalan, Motzkin, Delannoy and Schroder numbers, among many other well-studied sequences and arrays. Our main results give relationships between finiteness properties of the numbers πX(A), geometrical properties of the step set X, algebraic properties of the monoid \mathscr AX, 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.