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Skew Motzkin paths

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Abstract

In this paper, we study the class S of skew Motzkin paths, i.e., of those lattice paths that are in the first quadrat, which begin at the origin, end on the x-axis, consist of up steps U = (1, 1), down steps D = (1,−1), horizontal steps H = (1, 0), and left steps L = (−1,−1), and such that up steps never overlap with left steps. Let S n be the set of all skew Motzkin paths of length n and let s n = |S n |. Firstly we derive a counting formula, a recurrence and a convolution formula for sequence {s n } n ≥0. Then we present several involutions on S n and consider the number of their fixed points. Finally we consider the enumeration of some statistics on S n .

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, P. R. China

    Qing Lin Lu

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  1. Qing Lin Lu
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Correspondence to Qing Lin Lu.

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Supported by National Natural Science Foundation of China (Grant No. 11571150)

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Lu, Q.L. Skew Motzkin paths. Acta. Math. Sin.-English Ser. 33, 657–667 (2017). https://doi.org/10.1007/s10114-016-5292-y

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  • DOI: https://doi.org/10.1007/s10114-016-5292-y

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