Enumerations of Rational Non-decreasing Dyck Paths with Integer Slope

Abstract

We extend the concept of non-decreasing Dyck paths to t-Dyck paths. We denote the set of non-decreasing t-Dyck paths by ${{\mathcal D}}_t$. Several classic questions studied in other families of lattice paths are studied here for ${{\mathcal D}}_t$. We use generating functions, recursive relations and Riordan arrays to count, for example, the following aspects: the number of non-decreasing paths in ${{\mathcal D}}_t$ with a given fixed length, the total number of prefixes of all paths in ${{\mathcal D}}_t$ of a given length, and the total number of paths in ${{\mathcal D}}_t$ with a fixed number of peaks. We give a generating function to count the number of paths in ${{\mathcal D}}_t$ that can be written as a concatenation of a given fixed number of primitive paths and we give a relation between paths in ${{\mathcal D}}_t$ and direct column-convex polyominoes.