  This paper is a slightly extended version of the talk I gave at the Open
Problems in Algebraic Combinatorics conference at the University of Minnesota
in May 2022.
  We introduce two strict order relations on lattice paths and formulate
several open problems. The topic is related to Markov numbers, the Lagrange
spectrum, snake graphs and the cluster algebra of the once punctured torus.
  Our lattice paths are required to proceed by North and East steps and never
go over the diagonal. To define the order relations, we first construct a snake
graph $\mathcal{G}(\omega)$ and a band graph $\overline{\mathcal{G}(\omega)}$
for every such lattice path $\omega$. The first order relation $<_M$ is given
by the number of perfect matchings of the snake graphs. The second order
relation $<_L$ is given by the Lagrange number of a quadratic irrational
associated to the band graph.
