Abstract:

Given a point-lattice $(m+1) \times (n+1) \subseteq \mathbb{N} \times \mathbb{N}$ and $l \in \mathbb{N}$, we determine the number of royal paths from $(0,0)$ to $(m,n)$ with unit steps $(1,0)$, $(0,1)$ and $(1,1)$, which never go below the line $y = lx$, by means of the rotation principle. Compared to the method of ``penetrating analysis'', this principle has here the advantage of greater clarity and enables us to find meaningful additive decompositions of Schröder numbers. It also enables us to establish a connection to coordination numbers and the crystal ball in the cubic lattice $\mathbb{Z}^d$. As a by-product we derive a recursion for the number of North-East turns of rectangular lattice paths and construct a WZ-pair involving coordination numbers and Delannoy numbers.