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 T-path models from [43,44] for unpunctured surfaces to general surfaces. To do so, we introduce a new combinatorial way to generate these paths. The second is by using order ideals of labeled posets associated to arcs. In this context, we use the methods introduced in [25,26] 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.