Exceptional sequences and idempotent functions

Abstract

We prove that there is a one-to-one correspondence between the following three sets: idempotent functions on a set of size $n$, complete exceptional sequences of linear radical square zero Nakayama algebras of rank $n$ and rooted labeled forests with $n$ nodes and height of at most one. Therefore, the number of exceptional sequences is given by the sum ${\sum }_{j=1}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)j^{n-j}$.