Enumeration of the Motzkin paths above a line of rational slope

Abstract

We introduce Lm-Motzkin paths similar to m-Dyck paths. For fixed positive integer m, Lm-Motzkin paths are a generalization of Motzkin paths that start at (0,0), use steps U=(1,1), D=(1,−1) and L=(1,0), remain weakly above the line y=m−1mx, and end on this line. The number Mn(m) of Lm-Motzkin paths running from (0,0) to (mn,(m−1)n) is called the n-th Lm-Motzkin number. We first prove that the generating function Mm(t) of the Lm-Motzkin numbers satisfies the equation Mm(t)=1+tMm(t)m+t2Mm(t)2m. We then use this generating function and Riordan array to discuss the enumerations of the partial Lm-Motzkin paths and the partial grand Lm-Motzkin paths. Finally, we consider the colored Lm-Motzkin paths, and present two classes of γ-positive polynomials via enumerating these paths.