Enumeration of sharp peaks on Motzkin paths

Harold Yang

doi:10.1117/12.2686144

28 July 2023 Enumeration of sharp peaks on Motzkin paths

Harold Yang

Proceedings Volume 12756, 3rd International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2023); 127560A (2023) https://doi.org/10.1117/12.2686144
Event: 2023 3rd International Conference on Applied Mathematics, Modelling and Intelligent Computing (CAMMIC 2023), 2023, Tangshan, China

Abstract

Motzkin path is a type of lattice paths, which attract many combinatorists’ attention. In a Motzkin path, a node 𝑢 is called a sharp peak if it is between two steps 𝑈𝐷. A Motzkin path with no sharp peak is denoted by sp-free. In this paper, we focus on the distribution of the number of sharp peaks on Motzkin paths. In precise, by using Lagrange inversion formula, we obtain a formula to enumerate the number of Motzkin paths with length 𝑛 and containing 𝑘 sharp peaks. As the special case of the formula, we provide an elegant formula for the number of sp-free Motzkin paths.

Citation Download Citation

Harold Yang "Enumeration of sharp peaks on Motzkin paths", Proc. SPIE 12756, 3rd International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2023), 127560A (28 July 2023); https://doi.org/10.1117/12.2686144

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