  Electronic transport through chaotic quantum dots exhibits universal, system
independent, properties, consistent with random matrix theory. The quantum
transport can also be rooted, via the semiclassical approximation, in sums over
the classical scattering trajectories. Correlations between such trajectories
can be organized diagrammatically and have been shown to yield universal
answers for some observables. Here, we develop the general combinatorial
treatment of the semiclassical diagrams, through a connection to factorizations
of permutations. We show agreement between the semiclassical and random matrix
approaches to the moments of the transmission eigenvalues. The result is valid
for all moments to all orders of the expansion in inverse channel number for
all three main symmetry classes (with and without time reversal symmetry and
spin-orbit interaction) and extends to nonlinear statistics. This finally
explains the applicability of random matrix theory to chaotic quantum transport
in terms of the underlying dynamics as well as providing semiclassical access
to the probability density of the transmission eigenvalues.
