A Combinatorial Formula for Rank 2 Cluster Variables

Let $r$ be any positive integer, and let $x_1, x_2$ be indeterminates. We consider the sequence $\{x_n\}$ defined by the recursive relation $x_{n+1} =(x_n^r +1)/{x_{n-1}}$ for any integer $n$. Finding a combinatorial expression for $x_n$ as a rational function of $x_1$ and $x_2$ has been an open problem since 2001. We give a direct elementary formula for $x_n$ in terms of subpaths of a specific lattice path in the plane. The formula is manifestly positive, providing a new proof of a result by Nakajima and Qin.

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