Journal of Integer Sequences, Vol. 10 (2007), Article 07.7.7

Generalized Schröder Numbers and the Rotation Principle

Joachim Schröder
Universiteit van die Vrystaat
Departement van Wiskunde
Posbus 339
Bloemfontein 9300
South Africa


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.

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(Concerned with sequences A006318 A006319 A006320 A006321 A027307 A032349 A033296 A033877 A035597 A035598 A035599 A035600 A035601 A035602 A035603 A035604 A035605 A035606 A035607 and A106579 .)

Received January 8 2007; revised version received May 8 2007; July 25 2007. Published in Journal of Integer Sequences, July 25 2007.

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