A bijection between the sets of (a,b,b²)-Generalized Motzkin paths avoiding uvv-patterns and uvu-patterns

2022/04/17 by Yidong Sun, Cheng Sun, Sun, Yidong +3
Mathematics · Physics and Astronomy · #05A19 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Primary 05A15 #Secondary 05A10 #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2204.07906

openalex publication_date 2022/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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. Let τ be a word on \u, d, v, d\, denoted by G_n^τ(a,b,c) the set of τ-avoiding (a,b,c)-G-Motzkin paths of length n for a pattern τ. In this paper, we consider the uvv-avoiding (a,b,c)-G-Motzkin paths and provide a direct bijection σ between G_n^uvv(a,b,b²) and G_n^uvu(a,b,b²). Finally, the set of fixed points of σ is also described and counted.

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