Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises

2023/01/25 by Jean-Luc Baril, Baril, Jean-Luc, J. L. Ramírez +1
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2301.10449

openalex publication_date 2023/01/25 · openalex created_date 2023/02/13 · openalex updated_date 2026/07/28

Abstract

Motzkin paths with air pockets (MAP) of the first kind are defined as a generalization of Dyck paths with air pockets. They are lattice paths in ℕ² starting at the origin made of steps U=(1,1), D_k=(1,-k), k≥ 1 and H=(1,0), where two down-steps cannot be consecutive. We enumerate MAP and their prefixes avoiding peaks (resp. valleys, resp. double rise) according to the length, the type of the last step, and the height of its end-point. We express our results using Riordan arrays. Finally, we provide constructive bijections between these paths and restricted Dyck and Motzkin paths.

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