On the number of cycles in commutators of random permutations

We present general links between statistics of non-Hermitian random matrices and the distribution of the number of cycles of some specific random permutations. In particular, we derive explicit formulas for the generating functions of the number of cycles in the commutator $[\sigma,\tau] = \sigma \tau \sigma^{-1} \tau^{-1}$ where $\sigma$ is uniformly distributed, and $\tau$ is either one cycle, the product of two cycles of same size, or the product of many transpositions.