A Combinatorial Formula for Rank 2 Cluster Variables

TL;DR

This paper presents a new combinatorial formula for rank 2 cluster variables, expressing them as positive rational functions using lattice path substructures, solving an open problem since 2001.

Contribution

It provides an elementary, positive combinatorial formula for rank 2 cluster variables as rational functions of initial variables.

Findings

The formula is explicit and elementary.

It confirms positivity of cluster variables.

It offers a new proof of Nakajima and Qin's result.

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.