The combinatorics of frieze patterns and Markoff numbers

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).

Presented at the 18th International Conference on Formal Power Series and Algebraic Combinatorics. (Revised June 2006: I corrected a mis-statement at the end of section 2, and added reference to recent unpublished work of Hickerson. Revised May 2007: I correct a typo and added a paragraph.) Published as Integers, Volume 20 (2020), article A12; http://math.colgate.edu/~integers/u12/u12.pdf

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