  We introduce the notion of pattern in the context of lattice paths, and
investigate it in the specific case of Dyck paths. Similarly to the case of
permutations, the pattern-containment relation defines a poset structure on the
set of all Dyck paths, which we call the Dyck pattern poset. Given a Dyck path
P, we determine a formula for the number of Dyck paths covered by P, as well as
for the number of Dyck paths covering P. We then address some typical
pattern-avoidance issues, enumerating some classes of pattern-avoiding Dyck
paths. Finally, we offer a conjecture concerning the asymptotic behavior of the
sequence counting Dyck paths avoiding a generic pattern and we pose a series of
open problems regarding the structure of the Dyck pattern poset.
