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.

中文翻译：

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中山代数的二正则单模组合分类

Enomoto 证明对于有限维代数，有限生成射影模范畴上的精确结构的分类可以简化为 2-正则简单模的分类。在本文中，我们给出了 Nakayama 代数的 2-regular simple modules 的组合分类，我们使用这种分类来回答几个自然问题，例如在 Nakayama 代数的有限生成射影模的类别上何时存在独特的精确结构。我们还对 1-regular simple modules、准遗传中山代数和全局维度的中山代数进行了分类，最多两个。事实证明，大多数类都由众所周知的组合序列枚举，例如斐波那契数列、Riordan 数和 Narayana 数。