The combinatorics of frieze patterns and Markoff numbers

TL;DR

This paper introduces a combinatorial matchings model that explains the symmetries of frieze patterns, connects them to cluster algebras, and provides an enumerative interpretation of Markoff numbers with positive Laurent polynomial coefficients.

Contribution

It develops a novel combinatorial model linking frieze patterns, cluster algebras, and Markoff numbers, proving positivity of associated Laurent polynomials.

Findings

Matchings model explains frieze pattern symmetries

Enumerative interpretation of Markoff numbers established

Laurent polynomials associated with Markoff numbers have positive coefficients

Abstract

This article, based on joint work with Gabriel Carroll, Andy Itsara, Ian Le, Gregg Musiker, Gregory Price, Dylan Thurston, and Rui Viana, presents a combinatorial model based on perfect matchings that explains the symmetries of the numerical arrays that Conway and Coxeter dubbed frieze patterns. This matchings model is a combinatorial interpretation of Fomin and Zelevinsky's cluster algebras of type A. One can derive from the matchings model an enumerative meaning for the Markoff numbers, and prove that the associated Laurent polynomials have positive coefficients as was conjectured (much more generally) by Fomin and Zelevinsky. Most of this research was conducted under the auspices of REACH (Research Experiences in Algebraic Combinatorics at Harvard).