On Generalized Delannoy Paths

A Delannoy path is a minimal path with diagonal steps in ${\mathbb Z}^2$ between two arbitrary points. We extend this notion to the $n$ dimensions space ${\mathbb Z}^n$ and identify such paths with words on a special kind of alphabet: an S-alphabet. We show that the set of all the words corresponding to Delannoy paths going from one point to another is exactly one class in the congruence generated by a Thue system that we exhibit. This Thue system induces a partial order on this set that is isomorphic to the set of ordered partitions of a fixed multiset where the blocks are sets with a natural order relation. Our main result is that this poset is a lattice