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, \c{C}anak\c{c}{\i} 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 $\omega(a,b) = RR\cdots R UU \cdots U$ 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 $1+\sqrt5$.