Grand zigzag knight's paths

Abstract

We study the enumeration of different classes of grand knight's paths in the plane. In particular, we focus on the subsets of 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.

Figures (9)

• Figure 1: On the left: a grand knight's path of size 19, height 3, and altitude 1. On the right: a grand zigzag knight's path of size 19, height 1, and altitude $-2$.
• Figure 2: The eight grand knight's paths of size 4 ending on the $x$-axis.
• Figure 3: The six grand zigzag knight's paths of size 7 ending on the $x$-axis.
• Figure 4: The six grand zigzag knight's paths of size 10 having exactly the starting and the ending point on the $x$-axis.
• Figure 5: Illustration to Example 3.8. The path $\phi(X,Y)$ is a grand zigzag knight's path of size 27 and altitude 1.

Theorems & Definitions (47)

• Definition 1.1
• Definition 1.2
• Theorem 2.1
• Corollary 2.1
• Corollary 2.2
• Corollary 2.3
• Corollary 2.4
• Theorem 3.1
• Corollary 3.1
• Corollary 3.2