Lattice Walk Enumeration
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.
Cite
@article{arxiv.1803.10920,
title = {Lattice Walk Enumeration},
author = {Bryan Ek},
journal= {arXiv preprint arXiv:1803.10920},
year = {2018}
}
Comments
52 pages, 1 figure, 14 OEIS sequences (10 new). The paper and accompanying Maple package are contained <a href="http://sites.math.rutgers.edu/~bte14/Articles/ScoringPaths/ScoringPaths.html">here</a>. Edits from v3 are mostly cosmetic