On Two Orderings of Lattice Paths

Abstract

The \emph{Markov numbers} are positive integers appearing as solutions to the Diophantine equation $x^2+y^2+z^2=3xyz$. These numbers are very well-studied and have many combinatorial properties, as well as being the source of the long-standing unicity conjecture. In 2018, Çanakçı and Schiffler showed that the Markov number $m_{\frac{a}{b}}$ is the number of perfect matchings of a certain snake graph corresponding to the Christoffel path from $(0,0)$ to $(a,b)$. Based on this correspondence, Schiffler in 2023 introduced two orderings on lattice paths. For any path $\omega$, associate a snake graph $\mathcal{G}(\omega )$ and a continued fraction $g(\omega )$. The ordering ${<}_M$ is given by the number of perfect matchings on $\mathcal{G}(\omega )$, and the ordering ${<}_L$ is given by the Lagrange number of $g(\omega )$. In this work, we settle two conjectures of Schiffler. First, we show that the path $\omega (a,b)=RR\cdots RUU\cdots U$ is the unique maximum over all lattice paths from $(0,0)$ to $(a,b)$ with respect to both orderings ${<}_M$ and ${<}_L$. We then use this result to prove that $\sup L(\omega )$ over all lattice paths is exactly $1+\sqrt{5}$.

11 pages, 2 figures