Acta Mathematica Sinica, English Series

, Volume 33, Issue 5, pp 657–667 | Cite as

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 Sn be the set of all skew Motzkin paths of length n and let sn = |Sn|. Firstly we derive a counting formula, a recurrence and a convolution formula for sequence {sn}n≥0. Then we present several involutions on Sn and consider the number of their fixed points. Finally we consider the enumeration of some statistics on Sn.

Keywords

Dyck path Motzkin path skew Motzkin path enumeration 

MR(2010) Subject Classification

05A15 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouP. R. China

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