Some statistics on generalized Motzkin paths with vertical steps

Abstract

Recently, several authors have considered lattice paths with various steps, including vertical steps permitted. In this paper, we consider a kind of generalized Motzkin paths, called {\it G-Motzkin paths} for short, that is lattice paths from $(0,0)$ to $(n,0)$ in the first quadrant of the $XOY$-plane that consist of up steps $\mathbf{u}=(1,1)$, down steps $\mathbf{d}=(1,-1)$, horizontal steps $\mathbf{h}=(1,0)$ and vertical steps $\mathbf{v}=(0,-1)$. We mainly count the number of G-Motzkin paths of length $n$ with given number of $\mathbf{z}$-steps for $\mathbf{z}\in \{\mathbf{u},\mathbf{h},\mathbf{v},\mathbf{d}\}$, and enumerate the statistics "number of $\mathbf{z}$-steps" at given level in G-Motzkin paths for $\mathbf{z}\in \{\mathbf{u},\mathbf{h},\mathbf{v},\mathbf{d}\}$, some explicit formulas and combinatorial identities are given by bijective and algebraic methods, some enumerative results are linked with Riordan arrays according to the structure decompositions of G-Motzkin paths. We also discuss the statistics "number of ${\mathbf{z}}_1{\mathbf{z}}_2$-steps" in G-Motzkin paths for ${\mathbf{z}}_1,{\mathbf{z}}_2\in \{\mathbf{u},\mathbf{h},\mathbf{v},\mathbf{d}\}$, the exact counting formulas except for ${\mathbf{z}}_1{\mathbf{z}}_2=\mathbf{dd}$ are obtained by the Lagrange inversion formula and their generating functions.

37 pages, 3 figures