We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set $X\sub\Z^2$, 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~${A\in\A_X}$, 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 Schr\"oder numbers, among many other famous sequences and arrays. Our main results give the precise relationships between finiteness properties of the numbers $\pi_X(A)$, 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, and present a number of algorithms for computing the combinatorial data associated to finite step sets. Several examples are considered throughout to highlight the sometimes-subtle nature of the theoretical results.
Author(s): James East and Nicholas Ham
Publisher: arXiv
Year: 2018
Language: English
Pages: 63
Tags: lattice paths; enumeration; commutative monoids; 05A15; 05C38; 05C12; 20M14; 05C20; 05C30; 20M13; 05A10
1 Introduction......Page 1
2.1 Definitions and basic examples......Page 4
2.2 Finiteness properties: FPP, IPP and BPP......Page 7
2.3 Geometric conditions: CC, SLC and LC......Page 8
2.4 Recursion and further examples......Page 11
2.5 Small step sets......Page 16
2.6 Geometric, algebraic and combinatorial characterisations of the IPP......Page 17
2.7 An implicational hierarchy......Page 19
2.8 Groups......Page 21
2.9 Possible combinations of finiteness properties and geometric conditions......Page 25
2.10 Appendix (with a contribution from Stewart Wilcox): Combination (V)......Page 28
3.1 Definitions and basic examples......Page 34
3.2 Recursion and further examples......Page 36
3.3 Geometric conditions and finiteness properties for constrained walks......Page 41
3.4 Admissible steps, and constraint sets containing lattice cones......Page 44
4.1 Computing the points......Page 46
4.2 Checking the Line Condition......Page 49
4.3 Computing the numbers......Page 51
4.4 Further examples......Page 58
