Homological algebra of Nakayama algebras and 321-avoiding permutations

TL;DR

The paper studies linear Nakayama algebras in bijection with Dyck paths, which are themselves in bijection with $321$-avoiding permutations via the BJS map. It develops a homological interpretation of permutation data by translating Ext-type invariants into Dyck-path statistics, obtaining two main results: (i) for the Nakayama algebra $A_{\pi}$ associated to a $321$-avoiding permutation $\pi$, the number of indecomposable projective modules with injective dimension one equals the number of fixed points of $\pi$, and (ii) the self-extension space $\operatorname{Ext}^1_A(J,J)$ is isomorphic to $K^{\mathfrak{s}(\pi)}$, where $\mathfrak{s}(\pi)$ is the permutation’s support size. These findings yield enumerative corollaries: the number of linear Nakayama algebras with $n+2$ simples and a given Ext-dimension matches the number of standard tableaux of shape $[n,k]$, with maximal Ext-dimension attained in Catalan-number abundance; the work also provides explicit inverse correspondences between the combinatorial and algebraic data. The approach unifies homological algebra with classical combinatorics by leveraging the Dyck-path representation of Nakayama data and the BJS bijection, enabling precise combinatorial interpretations of homological invariants and guiding generalization to higher $k$ in future work.

Abstract

Linear Nakayama algebras over a field $K$ are in natural bijection to Dyck paths and Dyck paths are in natural bijection to 321-avoiding bijections via the Billey-Jockusch-Stanley bijection. Thus to every 321-avoiding permutation $π$ we can associate in a natural way a linear Nakayama algebra $A_π$. We give a homological interpretation of the fixed points statistic of 321-avoiding permutations using Nakayama algebras with a linear quiver. We furthermore show that the space of self-extension for the Jacobson radical of a linear Nakayama algebra $A_π$ is isomorphic to $K^{\mathfrak{s}(π)}$, where $\mathfrak{s}(π)$ is defined as the cardinality $k$ such that $π$ is the minimal product of transpositions of the form $s_i=(i,i+1)$ and $k$ is the number of distinct $s_i$ that appear.

Figures (6)

• Figure 1: An example of a Dyck $8$-path
• Figure 2: The points with level $k_i$
• Figure 3: An example of an indecomposable $A$-module
• Figure :
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Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.3
• Proposition 3.4
• proof
• Corollary 3.5