Linear Nakayama algebras over a field $K$ are in natural bijection to Dyck paths and Dyck paths are in natural bijection to 321-avoiding permutations 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-extensions 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.
@article{10_37236_13107,
author = {Eirini Chavli and Rene Marczinzik},
title = {Homological algebra of {Nakayama} algebras and 321-avoiding permutation},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {1},
doi = {10.37236/13107},
zbl = {8005220},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13107/}
}
TY - JOUR
AU - Eirini Chavli
AU - Rene Marczinzik
TI - Homological algebra of Nakayama algebras and 321-avoiding permutation
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/13107/
DO - 10.37236/13107
ID - 10_37236_13107
ER -
%0 Journal Article
%A Eirini Chavli
%A Rene Marczinzik
%T Homological algebra of Nakayama algebras and 321-avoiding permutation
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/13107/
%R 10.37236/13107
%F 10_37236_13107
Eirini Chavli; Rene Marczinzik. Homological algebra of Nakayama algebras and 321-avoiding permutation. The electronic journal of combinatorics, Tome 32 (2025) no. 1. doi: 10.37236/13107
