## Abstract

A k-generalized Dyck path of length n is a lattice path from (0, 0) to (n, 0) in the plane integer lattice Z × Z consisting of horizontal-steps (k, 0) for a given integer k ≥ 0, up-steps (1, 1), and down-steps (1, - 1), which never passes below the x-axis. The present paper studies three kinds of statistics on k-generalized Dyck paths: "number of u-segments", "number of internal u-segments" and "number of (u, h)-segments". The Lagrange inversion formula is used to represent the generating function for the number of k-generalized Dyck paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. Many important special cases are considered leading to several surprising observations. Moreover, enumeration results related to u-segments and (u, h)-segments are also established, which produce many new combinatorial identities, and specially, two new expressions for Catalan numbers.

Original language | English |
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Pages (from-to) | 2279-2292 |

Number of pages | 14 |

Journal | Discrete Applied Mathematics |

Volume | 156 |

Issue number | 12 |

DOIs | |

State | Published - 28 Jun 2008 |

### Bibliographical note

Funding Information:The authors are deeply grateful to the two anonymous referees for valuable suggestions on an earlier version of this paper which made it more readable. Thanks also to Simone Severini for helpful comments. The second author is supported by The National Science Foundation of China (10726021).

## Keywords

- Bell polynomials
- Catalan numbers
- Potential polynomials
- k-paths

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics