Every Markov triple has a unique q-deformed polynomial solution to the q-Markov equation, and these polynomials count weighted perfect matchings of snake graphs.
Higher $q$-Continued Fractions
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We introduce a $q$-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the $q$-rational numbers of Morier-Genoud and Ovsienko. They are defined as ratios of generating functions for $P$-partitions on certain posets. We give matrix formulas for computing them, which generalize previous results in the $q=1$ case. We also show that certain properties enjoyed by the $q$-rationals are also satisfied by our higher versions.
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On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges
Every Markov triple has a unique q-deformed polynomial solution to the q-Markov equation, and these polynomials count weighted perfect matchings of snake graphs.