Acta Mathematica Sinica, English Series

, Volume 33, Issue 5, pp 657–667

# Skew Motzkin paths

Article

## 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

05A15

## Preview

### References

1. [1]
Deutsch, E.: Dyck path enumeration. Discrete Math., 204, 167–202 (1999)
2. [2]
Deutsch, E., Munarini, E., Rinaldi, S.: Skew Dyck paths. J. Statist. Plann. Inference, 140, 2191–2203 (2010)
3. [3]
Deutsch, E., Munarini, E., Rinaldi, S.: Skew Dyck paths, area, and superdiagonal bargraphs. J. Statist. Plann. Inference, 140, 1550–1562 (2010)
4. [4]
Narayana, T. V.: A partial order and its application to probability. Sankhya, 21, 91–98 (1959)
5. [5]
Sapounakis, A., Tsikouras, P.: Counting peaks and valleys in k-colored Motzkin paths. Electron. J. Combin., 12, # R16 (2005)
6. [6]
Sedgewick, R., Flajolet, P.: An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA,1996
7. [7]
Sloane, N. J. A.: On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/~njas/sequences/Google Scholar
8. [8]
Wilf, H. S.: Generating Functionology, 2nd ed, Academic Press, New York, 1994Google Scholar

© 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