  We provide a combinatorial derivation of an asymptotic formula for averages
of mixed ratios of characteristic polynomials over the unitary group, where
mixed ratios are products of ratios and/or logarithmic derivatives. Our proof
of this formula is a generalization of Bump and Gamburd's elegant combinatorial
proof of Conrey, Forrester and Snaith's formula for averages of ratios of
characteristic polynomials over the unitary group. One application of this
formula is an asymptotic expression for sums over zeros of a random
characteristic polynomial from the unitary group, which we call an explicit
formula for eigenvalues in an analogy to what is called an explicit formula in
the context of $L$-functions.
