Knight's paths towards Catalan numbers

TL;DR

This paper explores the enumeration of partial knight's paths, establishing their connection to Catalan numbers and related combinatorial structures through algebraic proofs and bijections.

Contribution

It introduces new algebraic and combinatorial results linking knight's paths to Catalan, Motzkin, and Dyck paths, including bijections and enumeration formulas.

Findings

Zigzag knight's paths ending on the x-axis are counted by generalized Catalan numbers.

Constructive bijections are established with peakless Motzkin and Dyck paths.

Enumeration formulas for partial knight's paths are derived.

Abstract

We provide enumerating results for partial knight's paths of a given size. We prove algebraically that zigzag knight's paths of a given size ending on the $x$-axis are enumerated by the generalized Catalan numbers, and we give a constructive bijection with peakless Motzkin paths of a given length. After enumerating partial knight's paths of a given length, we prove that zigzag knight's paths of a given length ending on the $x$-axis are counted by the Catalan numbers. Finally, we give a constructive bijection with Dyck paths of a given length.