  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.
