  In this note we give a combinatorial and non-computational proof of the
asymptotics of the integer moments of the moments of the characteristic
polynomials of Haar distributed unitary matrices as the size of the matrix goes
to infinity. This is achieved by relating these quantities to a lattice point
count problem. Our main result is a new explicit expression for the leading
order coefficient in the asymptotic as a volume of a certain region involving
continuous Gelfand-Tsetlin patterns with constraints.
