Cluster expansion formulas and perfect matchings
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- Musiker, G. & Schiffler, R. J Algebr Comb (2010) 32: 187. doi:10.1007/s10801-009-0210-3
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We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph GT,γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph GT,γ.