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 $\sum\limits^n_{j=1}\binom{n}{j}j^{n-j}$.