In my recent work I have been led to consider the following type of permutation factorizations.

Let $\pi_1$ and $\pi_2$ be two fixed permutations, with disjoint support, i.e. $\pi_1$ acts on $\{1,...,n\}$ and $\pi_2$ acts on $\{n+1,...,n+m\}$.

Find all pairs $(\tau_1,\tau_2)$, such that $\pi_1\pi_2=\tau_1\tau_2$, with the condition that every cycle of the factors $\tau_1,\tau_2$ has exactly one element in $\{1,...,n\}$.

For example, $ (123)(45678)=(1 5)(2 4)(3 8 7 6)\cdot (1 4)(2 6 8)(3 5 7)$ is a case with $n=3$, $m=5$.

Has anyone seen a similar problem before? Does anyone have a suggestion for naming these things?