# Counting cycles after permuting within rows and columns

Consider a rectangular $p \times q$ array, labelled by the numbers $0, \ldots, pq - 1$ for convenience. Let $S_p$ and $S_q$ and $S_{pq}$ denote the symmetric groups. Take a family of permutations:

$$\rho_1, \ldots, \rho_q \in S_p,$$ $$\kappa_1, \ldots, \kappa_p \in S_q.$$

Then letting $\rho_t$ act on the $t^{th}$ row, one obtains a permutation $\rho \in S_{pq}$ which leaves the rows of the array stable (but permutes within each row). Similarly, one obtains a permutation $\kappa \in S_{pq}$ which leaves the columns of the array stable.

Can one say anything (beyond parity) that relates the cycle structure (say, the number of cycles) of $\rho \circ \kappa$ to the cycle structure of $\rho$ and $\kappa$ separately?

What if the cycle structure of every $\rho_i$ is the same, and the cycle structure of every $\kappa_j$ is the same?