  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.
