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Enomoto showed for finite dimensional algebras that the classification ofexact structures on the category of finitely generated projective modules canbe reduced to the classification of 2-regular simple modules. In this article,we give a combinatorial classification of 2-regular simple modules for Nakayamaalgebras and we use this classification to answer several natural questionssuch as when there is a unique exact structure on the category of finitelygenerated projective modules for Nakayama algebras. We also classify 1-regularsimple modules, quasi-hereditary Nakayama algebras and Nakayama algebras ofglobal dimension at most two. It turns out that most classes are enumerated bywell-known combinatorial sequences, such as Fibonacci, Riordan and Narayananumbers. We first obtain interpretations in terms of the Auslander-Reitenquiver of the algebra using homological algebra, and then apply suitablebijections to relate these to combinatorial statistics on Dyck paths.

A combinatorial classification of 2-regular simple modules for Nakayama algebras