The number of elements in the mutation class of a quiver of type $D_n$

TL;DR

This paper derives a formula for counting the number of quivers in the mutation class of type D_n using a correspondence with rooted trees, providing a combinatorial approach to a problem in cluster algebra theory.

Contribution

It introduces a novel combinatorial method linking quivers in mutation classes to rooted trees, leading to an explicit counting formula for type D_n quivers.

Findings

Derived an explicit formula for the number of quivers in the mutation class of type D_n

Established a correspondence between quivers and rooted trees

Provided a combinatorial proof for the enumeration result

Abstract

We show that the number of quivers in the mutation class of a quiver of Dynkin type $D_n$ is given by $\sum_{d|n} \phi(n/d)\binom{2d}{d}/(2n)$ for $n \geq 5$. To obtain this formula, we give a correspondence between the quivers in the mutation class and certain rooted trees.