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 Tpath 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. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://