Homological algebra of Nakayama algebras and 321-avoiding permutations

TL;DR

This paper explores the deep connections between Nakayama algebras, Dyck paths, and 321-avoiding permutations, providing a homological interpretation of permutation statistics through algebraic structures.

Contribution

It introduces a novel homological perspective linking 321-avoiding permutations with Nakayama algebras, including interpretations of fixed points and self-extension spaces.

Findings

Homological interpretation of fixed points in 321-avoiding permutations.

Isomorphism between self-extension space of Jacobson radical and a vector space over K.

Bijections connecting Nakayama algebras, Dyck paths, and 321-avoiding permutations.

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 $\pi$ we can associate in a natural way a linear Nakayama algebra $A_{\pi}$. 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 $A_{\pi}$ is isomorphic to $K^{\mathfrak{s}(\pi)}$, where $\mathfrak{s}(\pi)$ is defined as the cardinality $k$ such that $\pi$ 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.