Enumerating Restricted Dyck Paths with Context-Free Grammars

TL;DR

This paper explores how context-free grammars can be used to enumerate restricted Dyck paths, extending previous work by Zeilberger to more complex families and deriving new combinatorial results.

Contribution

It generalizes Zeilberger's grammars to infinite families and highlights combinatorial results obtained through grammatical proofs.

Findings

Derived new generating function identities for restricted Dyck paths

Generalized grammars to infinite families of Dyck paths

Provided combinatorial proofs using context-free grammars

Abstract

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 and generalizing some of Zeilberger's grammars to infinite families.