Lattice Walk Enumeration

TL;DR

This paper introduces a Maple package for enumerating 2D lattice walks with various restrictions, providing explicit solutions or functional descriptions for their generating functions, aiding combinatorial analysis.

Contribution

It presents a systematic computational approach to derive generating functions for lattice walks with arbitrary step sets, including explicit rational and algebraic solutions.

Findings

Explicit rational solutions for bounded walks.

Algebraic solutions for semi-bounded or unbounded walks.

A self-referential description for complex cases.

Abstract

Trying to enumerate all of the walks in a 2D lattice is a fun combinatorial problem and there are numerous applications, from polymers to sports. Computers provide a wonderful tool for analyzing these walks; we provide a Maple package for automatically describing generating functions of walks restricted to any step set in a 2D lattice. We always obtain a closed system of relations for generating functions of walks that are bounded, semi-bounded, or unbounded. For bounded walks, this leads to explicit rational solutions! For semi-bounded or unbounded walks, we may get lucky and obtain algebraic solutions; if not, we still have a short self-referential description of the generating function.