Grand zigzag Knight's paths
Universit\'e de Bourgogne, Jean-Luc Baril, Nathanaël Hassler, Universit\'e de Bourgogne, Sergey Kirgizov, Universit\'e de Bourgogne, Jos\'e L. Ramırez, Universidad Nacional de Colombia · Enumerative Combinatorics and Applications · 2024
We study the enumeration of different classes of grand knight's paths in the plane.In particular, we focus on the subsets of the zigzag knight's paths that are subject to constraints.These constraints include ending at y-coordinate 0, bounded by a horizontal line, confined within a tube, among other considerations.We present our results using generating functions or direct closed-form expressions.We derive asymptotic results, finding approximations for quantities such as the probability that a zigzag knight's path stays in some area of the plane, or for the average of the altitude of such a path.Additionally, we exhibit some bijections between grand zigzag knight's paths and some pairs of compositions.