Abstract

Given a point-lattice (m + 1) × (n + 1) ⊆ ℕ × ℕ and l ∈ 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 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.