  The statistics of quantum transport through chaotic cavities with two leads
is encoded in transport moments $M_m={\rm Tr}[(t^\dag t)^m]$, where $t$ is the
transmission matrix, which have a known universal expression for systems
without time-reversal symmetry. We present a semiclassical derivation of this
universality, based on action correlations that exist between sets of long
scattering trajectories. Our semiclassical formula for $M_m$ holds for all
values of $m$ and arbitrary number of open channels. This is achieved by
mapping the problem into two independent combinatorial problems, one involving
pairs of set partitions and the other involving factorizations in the symmetric
group.
