2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169287We study cluster algebras with principal and arbitrary 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 certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincaré characteristic of quiver Grassmannians in Dynkin type $A$ and affine Dynkin type $\tilde A$.36 pages, 9 figuresRepresentation TheoryRings and Algebras16S99; 05E99; 16G20text