  Our results can be viewed as applications of algebraic combinatorics in
random matrix theory. These applications are motivated by the predictive power
of random matrix theory for the statistical behavior of the celebrated Riemann
$\zeta$-function (and $L$-functions in general), which was discovered by
Montgomery (with regard to zeros of $L$-functions) and by Keating and Snaith
(with regard to values of $L$-functions).
  The first results revolve around a new operation on partitions, which we call
overlap. We prove two overlap identities for so-called Littlewood-Schur
functions. The first overlap identity represents the Littlewood-Schur function
$LS _\lambda(X; Y)$ as a sum over subsets of $X$, while the second overlap
identity essentially represents $LS_\lambda(X; Y)$ as a sum over pairs of
partitions whose overlap equals $\lambda$. Both identities are derived by
applying Laplace expansion to a determinantal formula for Littlewood-Schur
functions due to Moens and Van der Jeugt. In addition, we give two visual
characterizations for the set of all pairs of partitions whose overlap is equal
to a partition $\lambda$.
  The second result is 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. The generalization relies on three
combinatorial results, namely the first overlap identity, a new variant of the
Murnaghan-Nakayama rule and an idea from vertex operator formalism. We conclude
this thesis by explaining how this approach might lead to new number theoretic
proofs.
