# Enumerating Restricted Dyck Paths with Context-Free Grammars.

@article{Bu2020EnumeratingRD, title={Enumerating Restricted Dyck Paths with Context-Free Grammars.}, author={AJ Bu and Robert Dougherty-Bliss}, journal={arXiv: Combinatorics}, year={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 grammar. In a recent paper, Zeilberger showed that many restricted sets of Dyck paths satisfy different, more complicated grammars, and from this derived various generating function identities. We take this further, highlighting some combinatorial results about Dyck paths obtained via grammatical proof…

## One Citation

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## References

SHOWING 1-7 OF 7 REFERENCES

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

- Mathematics
- 2003

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.…

### Automatic Counting of Restricted Dyck Paths via (Numeric and Symbolic) Dynamic Programming

- Mathematics
- 2020

Dyck paths are one of the most important objects in enumerative combinatorics, and there are many papers devoted to counting selected families of Dyck paths. Here we present two approaches for the…

### The On-Line Encyclopedia of Integer Sequences

- Computer ScienceElectron. J. Comb.
- 1994

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.

### Motzkin Numbers

- MathematicsEur. J. Comb.
- 1998

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

### Dyck Paths With No Peaks At Height k

- Mathematics
- 2001

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…

### Analytic Combinatorics

- Mathematics
- 2009

This text can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study, and is certain to become the definitive reference on the topic.