Enumerating 2D and 3D lattice paths with arbitrary steps
Cemil Karaçam, Alper Vural · Mathematica Bohemica · 2025
Let $S$ be a finite set of integer vectors. We consider lattice paths that use only the vectors in $S$. We focus on paths that use a fixed number of vectors. We generally assume vectors in $S$ have a fixed coordinate sum, which allows us to determine the number of vectors in a path, which we call its length. We count the number of paths with fixed length for various sets of vectors $S$. We then use our enumeration results to determine the minimal length path given a terminal point. First, we explore this problem in $S \subseteq\mathbb{N}^3$. After solving the problem of enumeration and determining the minimal length for various sets $S\subseteq\mathbb{N}^3$, we solve these problems for a general case $S=\{(1,0),(0,1),(u,v),(v,u)\}$. We conclude with an enumeration problem of paths that stay weakly below the line $y=x$.