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A Relation Between Schröder Paths and Motzkin Paths

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Abstract

A small q-Schröder path of semilength n is a lattice path from (0, 0) to (2n, 0) using up steps \(U = (1, 1)\), horizontal steps \(H = (2, 0)\), and down steps \(D = (1,-1)\) such that it stays weakly above the x-axis, has no horizontal steps on the x-axis, and each horizontal step comes in q colors. In this paper, we provide a bijection between the set of small q-Schröder paths of semilength \(n+1\) and the set of \((q+2, q+1)\)-Motzkin paths of length n. Furthermore, a one-to-one correspondence between the set of small 3-Schröder paths of semilength n and the set of Catalan rook paths of semilength n is obtained, and a bijection between small 4-Schröder paths and Catalan queen paths is also presented.

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Acknowledgements

The authors would like to thank the referees for their helpful suggestions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11861045, 11561044) and the Hongliu Foundation of First-class Disciplines of Lanzhou University of Technology, China.

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  1. Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, 730050, Gansu, People’s Republic of China

    Lin Yang & Sheng-Liang Yang

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  1. Lin Yang
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  2. Sheng-Liang Yang
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Correspondence to Lin Yang.

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Yang, L., Yang, SL. A Relation Between Schröder Paths and Motzkin Paths. Graphs and Combinatorics 36, 1489–1502 (2020). https://doi.org/10.1007/s00373-020-02185-6

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  • DOI: https://doi.org/10.1007/s00373-020-02185-6

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