  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.
