# Dyck Paths with Peaks Avoiding or Restricted to a Given Set

@article{Eu2003DyckPW, title={Dyck Paths with Peaks Avoiding or Restricted to a Given Set}, author={Sen-Peng Eu and Shu-Chung Liu and Yeong-Nan Yeh}, journal={Studies in Applied Mathematics}, year={2003}, volume={111} }

In this paper we focus on Dyck paths with peaks avoiding or restricted to an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We also discuss a special case that requires all peak heights to either lie on or avoid a congruence class (or classes) modulo k. The case when k= 2 is especially interesting. The two sequences for this case are proved, combinatorially as well as algebraically, to be the Motzkin numbers and the…

## 22 Citations

### Dyck Paths with Peak‐ and Valley‐Avoiding Sets

- Mathematics
- 2008

In this paper, we focus on Dyck paths with peaks and valleys, avoiding an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We…

### Peak-and Valley-Avoiding Sets

- Mathematics
- 2008

In this paper, we focus on Dyck paths with peaks and valleys, avoiding an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We…

### On Hankel determinants for Dyck paths with peaks avoiding multiple classes of heights

- MathematicsEur. J. Comb.
- 2022

### Proof of a conjecture on Hankel determinants for Dyck paths with restricted peak heights

- MathematicsDiscrete Mathematics
- 2022

### Enumerating Restricted Dyck Paths with Context-Free Grammars.

- Mathematics
- 2020

The number of Dyck paths of semilength $n$ is famously $C_n$, the $n$th Catalan number. This fact follows after noticing that every Dyck path can be uniquely parsed according to a context-free…

### General Results on the Enumeration of Strings in Dyck Paths

- MathematicsElectron. J. Comb.
- 2011

In this paper, arbitrary strings are considered, and the statistic "number of occurrences of $\tau$ at height at least $j$" is considered, evaluating the corresponding generating function when $\t Tau$ is either a Dyck prefix or a Dycki suffix.

### Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects

- Mathematics, Computer ScienceLATA
- 2018

This article presents a powerful method for the enumeration of pattern-avoiding words generated by an automaton or a context-free grammar. It relies on methods of analytic combinatorics, and on a…

### #A69 INTEGERS 21 (2021) ENUMERATING RESTRICTED DYCK PATHS WITH CONTEXT FREE GRAMMARS

- Mathematics
- 2021

The number of Dyck paths of semilength n is famously Cn, the nth Catalan number. This fact follows after noticing that every Dyck path can be uniquely parsed according to a context-free grammar. In a…

### Analytic Combinatorics of Lattice Paths with Forbidden Patterns, the Vectorial Kernel Method, and Generating Functions for Pushdown Automata

- Mathematics, Computer ScienceAlgorithmica
- 2019

A vectorial kernel method is developed which solves in a unified framework all the problems related to the enumeration of words generated by a pushdown automaton for the enumerations of lattice paths that avoid a fixed word (a pattern), or for counting the occurrences of a given pattern.

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