Homological algebra of Nakayama algebras 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 $π$ 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.