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

Rene Marczinzik, Martin Rubey, Christian Stump

Published 2018-11-14Version 1

We discuss combinatorial interpretations of several homological properties of Nakayama algebras in terms of Dyck path statistics. We thereby classify and enumerate various classes of Nakayama algebras. Most importantly, we classify their 2-regular simple modules, corresponding to exact structures on the category of projective modules. We also classify 1-regular simple modules, quasi-hereditary Nakayama algebras and Nakayama algebras of global dimension at most two. As it turns out, 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. Then, we apply suitable bijections to relate these to combinatorial statistics on Dyck paths.