Coxeter's Frieze Patterns Arising from Dyck Paths

TL;DR

This paper introduces a new approach to constructing integral frieze patterns using Dyck paths and diamonds of Dynkin type An, linking combinatorial objects with algebraic structures.

Contribution

It defines the notion of diamonds of Dynkin type An and demonstrates their use in generating frieze patterns from Dyck paths, providing a novel combinatorial framework.

Findings

Diamonds of Dynkin type An effectively generate frieze patterns.

A new combinatorial method connects Dyck paths with algebraic frieze patterns.

The approach broadens understanding of frieze patterns through combinatorial structures.

Abstract

Frieze patterns are defined by objects of a category of Dyck paths, to do that, it is introduced the notion of diamond of Dynkin type An. Such diamonds constitute a tool to build integral frieze patterns.