  After Gross, Hacking, Keel, Kontsevich [GHKK18] introduced the theta basis
which is shown to be indexed by its highest term exponent in cluster variables
of any given seed, we are interested in all the non-vanishing exponents in
these cluster variables. We prove that the coefficients of the exponents of any
cluster variable of type $A_n$ are log-concave. We show that the cluster
monomials of $A_2$ type are log-concave. As for larger generality, we
conjecture that the log-concavity of cluster monomials is also true.
