Log-concavity of cluster algebras of type $A_n$

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.
Comment: 26 pages, 10 figures