A Combinatorial Formula for Rank 2 Cluster Variables

Abstract

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.

17 pages, v2:a corollary to the main theorem added, v3:another corollary added

References in corpus (2)

• Positivity for cluster algebras from surfaces
• On cluster variables of rank two acyclic cluster algebras