Exceptional Sequences and Idempotent Functions

TL;DR

This paper establishes a bijective correspondence between idempotent functions, exceptional sequences of certain Nakayama algebras, and rooted labeled forests, providing a combinatorial enumeration of exceptional sequences.

Contribution

It introduces a novel correspondence linking algebraic structures with combinatorial objects, leading to an explicit formula for counting exceptional sequences.

Findings

One-to-one correspondence between idempotent functions, exceptional sequences, and rooted forests.

Derived a closed-form enumeration formula for exceptional sequences.

Connected algebraic and combinatorial structures in representation theory.

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