  In this paper, we introduce and analyze a new operation $\circ_{R}$ which
mixes two distributions $\Omega$ and $\Omega'$ via a random orthogonal matrix.
In particular, we take $\Omega \circ_R \Omega'$ to be the limit as $n \to
\infty$ of the distribution of singular values of $DRD'$ where $D$ and $D'$ are
$n \times n$ diagonal matrices whose diagonal entries have distributions
$\Omega$ and $\Omega'$ respectively and $R$ is a random $n \times n$ orthogonal
matrix. We show that $\circ_R$ has several nice properties. We first observe
that $\circ_R$ is commutative and associative and compute the moments of
$\Omega \circ_R \Omega'$ in terms of the moments of $\Omega$ and $\Omega'$. We
then show that $\circ_R$ interacts very nicely with the spectrum of the
singular values of Z-shaped and multi-Z-shaped graph matrices. This allows us
to answer the question posed by our previous paper of how to describe the
spectrum of the singular values of Z-shaped and multi-Z-shaped graph matrices
when the input distribution is not $\{-1,1\}$. In our analysis, we show that
the moments of our distributions are closely connected to non-crossing
partitions and prove a number of new results on non-crossing partitions which
may be of independent interest.
