The uvu-Avoiding (a, b, c)-Generalized Motzkin Paths with Vertical Steps: Bijections and Statistic Enumerations

A generalized Motzkin path, called G-Motzkin path for short, of length n is a lattice path from (0, 0) to (n, 0) in the first quadrant of the XOY-plane that consists of up steps u = (1, 1), down steps d = (1, -1), horizontal steps h = (1, 0) and vertical steps v = (0, -1). An (a, b, c)-G-Motzkin path is a weighted G-Motzkin path such that the u-steps, h-steps, v-steps and d-steps are weighted respectively by 1, a, b and c. In this paper, we first give bijections between the set of uvu-avoiding (a, b, b(2))-G-Motzkin paths of length n and the set of (a, b)-Schroder paths as well as the set of (a + b, b)-Dyck paths of length 2n, between the set of {uvu, uu}-avoiding (a, b, b(2))- G-Motzkin paths of length n and the set of (a + b, ab)-Motzkin paths of length n, between the set of {uvu, uu}-avoiding (a, b, b(2))-G-Motzkin paths of length n + 1 beginning with an h-step weighted by a and the set of (a, b)-Dyck paths of length 2n+2. In the last section, we focus on the enumeration of statistics "number of z-steps" for z ? {u, h, v, d} and "number of points" at given level in uvu-avoiding G-Motzkin paths. These counting results are linked with Riordan arrays.