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Combinatorial problems in the semiclassical approach to quantum chaotic transport

A semiclassical approach to the calculation of transport moments $M_m={\rm Tr}[(t^†t)^m]$, where $t$ is the transmission matrix, was developed in [M. Novaes, Europhys. Lett. 98, 20006 (2012)] for chaotic cavities with two leads and broken time-reversal symmetry. The result is an expression for $M_m$ as a perturbation series in 1/N, where N is the total number of open channels, which is in agreement with random matrix theory predictions. The coefficients in this series were related to two open combinatorial problems. Here we expand on this work, including the solution to one of the combinatorial problems. As a by-product, we also present a conjecture relating two kinds of factorizations of permutations.