## Abstract

A Dyck path of length 2n is a lattice path from (0, 0) to (2n, 0) consisting of upsteps u = (1, 1) and down-steps d = (1,-1) which never passes below the x-axis. Let D _{n} denote the set of Dyck paths of length 2n. A peak is an occurrence of ud (an upstep immediately followed by a downstep) within a Dyck path, while a valley is an occurrence of du. Here, we compute explicit formulas for the generating functions which count the members of D _{n} according to the maximum number of steps between any two peaks, any two valleys, or a peak and a valley. In addition, we provide closed expressions for the total value of the corresponding statistics taken over all of the members of D _{n}. Equivalent statistics on the set of 231-avoiding permutations of length n are also described.

Original language | English |
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Article number | 12.1.1 |

Journal | Journal of Integer Sequences |

Volume | 15 |

Issue number | 1 |

State | Published - 15 Dec 2011 |

## Keywords

- Dyck paths
- Generating functions
- Peaks
- Statistics
- Valleys

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics