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\begin{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
Schr\"oder 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.
\end{abstract}

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