  Enomoto showed for finite dimensional algebras that the classification of
exact structures on the category of finitely generated projective modules can
be reduced to the classification of 2-regular simple modules. In this article,
we give a combinatorial classification of 2-regular simple modules for Nakayama
algebras and we use this classification to answer several natural questions
such as when there is a unique exact structure on the category of finitely
generated projective modules for Nakayama algebras. We also classify 1-regular
simple modules, quasi-hereditary Nakayama algebras and Nakayama algebras of
global dimension at most two. It turns out that most classes are enumerated by
well-known combinatorial sequences, such as Fibonacci, Riordan and Narayana
numbers. We first obtain interpretations in terms of the Auslander-Reiten
quiver of the algebra using homological algebra, and then apply suitable
bijections to relate these to combinatorial statistics on Dyck paths.
