On Cluster Variables of Rank Two Acyclic Cluster Algebras

In this note, we find an explicit formula for the Laurent expression of cluster variables of coefficient-free rank two cluster algebras associated with the matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\left(\begin{array}{ll} 0 &c \\ -c &0 \end{array}\right)}}$$\end{document}, which manifestly shows that a large number of coefficients are non-negative. As a corollary, we obtain an explicit expression for the Euler-Poincaré characteristics of the corresponding quiver Grassmannians.