Geodesic
Cluster expansion formulas and perfect matchings
Journal of Algebraic Combinatorics, Tome 32 (2010) no. 2, pp. 187-209.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $G _{ T, \gamma }$ 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 $G _{ T, \gamma }$.
Keywords: cluster algebra, triangulated surface, principal coefficients, F-polynomial, snake graph 
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     title = {Cluster expansion formulas and perfect matchings},
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Musiker, Gregg; Schiffler, Ralf. Cluster expansion formulas and perfect matchings. Journal of Algebraic Combinatorics, Tome 32 (2010) no. 2, pp. 187-209. http://geodesic.mathdoc.fr/item/JAC_2010__32_2_a5/