Two Formulas for F-Polynomials

We discuss a product formula for F-polynomials in cluster algebras and provide two proofs. One proof is inductive and uses only the mutation rule for F-polynomials. The other is based on the Fock–Goncharov decomposition of mutations. We conclude by expanding this product formula as a sum and illustrate applications. This expansion provides an explicit combinatorial computation of F-polynomials in a given seed that depends only on the c-vectors and g-vectors along a finite sequence of mutations from the initial seed to the given seed.