Cluster algebras and continued fractions

Understand

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets.

• These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras.
• To a continued fraction $[a_1,a_2,\ldots,a_n]$, we associate a snake graph $\mathcal{G}[a_1,a_2,\ldots,a_n]$ such that the continued fraction is the quotient of the number of perfect matchings of $\mathcal{G}[a_1,a_2,\ldots,a_n]$ and $\mathcal{G}[a_2,\ldots,a_n]$.
• We also show that snake graphs are in bijection with continued fractions.