Enumerations of peaks and valleys on non-decreasing Dyck paths

A valley in a Dyck path is a local minimum, and a peak is a local maximum. A Dyck path is non-decreasing if the y-coordinates of the valleys of the path valley form anon-decreasing sequence. In this paper we provide some statistics about peaks and valleys in non-decreasing Dyck paths, such as their total number, the number of low and high valleys, low and high peaks, etc. Our methods include bijective proofs, recursive relations, and the symbolic method for generating functions.