Mixed cobinary trees

TL;DR

This paper introduces mixed cobinary trees, a variation of binary trees, and explores their connection to cluster theory, quivers, and generalized associahedra, revealing their enumeration by Catalan numbers and their role in representation theory.

Contribution

It presents a new combinatorial structure called mixed cobinary trees and establishes their relation to cluster theory, quivers, and generalized associahedra, extending known results.

Findings

Number of isomorphism classes equals Catalan number Cn.

Established bijection between mixed cobinary trees and vertices of generalized associahedron.

Connected mixed cobinary trees to quiver representations and c-vectors.

Abstract

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees. We show that the number of isomorphism classes of such trees is given by the Catalan number Cn where n is the number of internal nodes. We also consider the corresponding quiver Q_{\epsilon} of type An-1. As a special case of more general known results about the relation between c-vectors, representations of quivers and their semi-invariants, we explain the bijection between mixed cobinary trees and the vertices of the generalized associahedron corresponding to the quiver Q_{\epsilon}.