Cluster Expansions: T-walks, Labeled Posets and Matrix Calculations

We give two new combinatorial methods for computing cluster expansion formulas for arcs coming from possibly punctured surfaces. The first is by using $T$-walks, an extension of the $T$-path model for unpunctured surfaces to general surfaces. We also introduce a new way of generating $T$-paths. The second method is by using order ideals of labeled posets associated to arcs. We also use the theory of oriented posets to give a quick way to calculate the expressions using $2$ by $2$ matrices. The techniques introduced are applicable to different settings in cluster algebras and beyond.