Quiddities of polygon dissections and the Conway-Coxeter frieze equation

TL;DR

This paper explores a matrix equation linked to Coxeter frieze patterns, connecting group theory, continued fractions, and combinatorics, and extends classical results by identifying new integer sequences including Catalan numbers.

Contribution

It introduces a new matrix equation related to Coxeter frieze patterns, counts its positive solutions, and uncovers new integer sequences extending classical combinatorial results.

Findings

Derived a series of integer sequences from the matrix equation

Connected solutions to classical Catalan numbers

Identified new sequences related to Coxeter frieze patterns

Abstract

We study a $2 \times 2$ matrix equation arising naturally in the theory of Coxeter frieze patterns. It is formulated in terms of the generators of the group $\mathrm{PSL}(2,\mathbb{Z})$ and is closely related to continued fractions. It appears in a number of different areas, for example, toric varieties. We count its positive solutions, obtaining a series of integer sequences, some known and some new. This extends classical work of Conway and Coxeter proving that the first of these sequences is the Catalan numbers.