  We prove a simple formula for arbitrary cluster variables in the marked
surfaces model. As part of the formula, we associate a labeled poset to each
tagged arc, such that the associated $F$-polynomial is a weighted sum of order
ideals. Each element of the poset has a weight, and the weight of an ideal is
the product of the weights of the elements of the ideal. In the unpunctured
case, the weight on each element is a single $\hat{y}_i$, in the usual sense of
principal coefficients. In the presence of punctures, some elements may have
weights of the form $\hat{y}_i/\hat{y}_j$. Our search for such a formula was
inspired by the Fundamental Theorem of Finite Distributive Lattices combined
with work of Gregg Musiker, Ralf Schiffler, and Lauren Williams that, in some
cases, organized the terms of the $F$-polynomial into a distributive lattice.
The proof consists of a simple and poset-theoretically natural argument in a
special case, followed by a hyperbolic geometry argument using a cover of the
surface to prove the general case.
