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… 

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References

SHOWING 1-10 OF 21 REFERENCES

Permutations with Restricted Patterns and Dyck Paths

TLDR
Using this bijection, it is shown that all the recently discovered results on generating functions for 132-avoiding permutations with a given number of occurrences of the pattern 12...k follow directly from old results on the enumeration of Motzkin paths, among which is a continued fraction result due to Flajolet.

Counting peaks at height k in a Dyck path

A Dyck path is a lattice path in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of steps (1,1) and (1,-1), which never passes below the x-axis. A peak at height k on a Dyck path is

Combinatorial aspects of continued fractions

Dyck Paths With No Peaks At Height k

A Dyck path of length 2n is a path in two-space from (0, 0) to (2n, 0) which uses only steps (1, 1) (north-east) and (1,−1) (south-east). Further, a Dyck path does not go below the x-axis. A peak on

A survey of the Fine numbers

Motzkin Numbers

TLDR
Motzkin numbers (which are related to Catalan numbers) are studied and it is shown that the sequenceMnis logarithmically concave with limMn+1/Mn=3.

The On-Line Encyclopedia of Integer Sequences

  • N. Sloane
  • Computer Science
    Electron. J. Comb.
  • 1994
TLDR
The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.

Dyck path enumeration

An introduction to the analysis of algorithms

TLDR
This chapter discusses methods for Solving Recurrences of Trees, Representations of Trees and Binary Trees, and Analyzing Properties of Permutations with CGFs.