We present an initial-seed-mutation formula for
-vectors
of cluster variables in a cluster algebra. We also give two
rephrasings of this recursion: one as a duality formula for
-vectors in the style of the
-vectors/-vectors
dualities of Nakanishi and Zelevinsky, and one as a formula expressing the
highest powers in the Laurent expansion of a cluster variable in terms of the
-vectors
of any cluster containing it. We prove that the initial-seed-mutation recursion holds
in a varied collection of cluster algebras, but not in general. We conjecture
further that the formula holds
for source-sink moves on the initial seed in
an arbitrary cluster algebra, and we prove this conjecture in the case of
surfaces.
