Coloured quivers of type A and the cell-growth problem

TL;DR

This paper explores the enumeration of coloured quivers in the context of $m$-cluster categories of Dynkin type A, linking geometric polygon angulations to the cell-growth problem, and providing new combinatorial insights.

Contribution

It introduces a geometric approach to count coloured quivers in $m$-mutation classes of type A and connects this to polygon angulations and the cell-growth problem.

Findings

Count of coloured quivers in $m$-mutation class of type A

Connection between polygon angulations and quiver enumeration

Application to the cell-growth problem

Abstract

We use a geometric description of $m$-cluster categories of Dynkin type $A$ to count the the number of coloured quivers in the $m$-mutation class of quivers of Dynkin type $A$. This is related to angulations of polygons and the cell-growth problem.