2022/01/23 by Yidong Sun, Weichen Wang, Sun, Yidong +3 · 1 citation
Computer Science · Mathematics · #05A19 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Primary 05A15 #Secondary 05A10 #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2201.09236
openalex publication_date 2022/01/23 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
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,b2)-G-Motzkin paths of length n and the set of (a,b)-Schröder paths as well as the set of (a+b,b)-Dyck paths of length 2n, between the set of \uvu, uu\-avoiding (a,b,b2)-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,b2)-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.