Let $\pi$ be a permutation in $S_n$ with cycle type $\lambda$.

How many factorizations into two factors $\pi=\sigma_1\sigma_2$ are there, such that the subgroup $\langle \sigma_1,\sigma_2\rangle$ generated by $\sigma_1$ and $\sigma_2$ has $k$ orbits?

(I impose no condition at all on the factors.)

For $k=1$, I am asking for the total number of transitive factorizations of $\pi$ into two factors. At least this case should be known, I hope.

I present the numerical results, as example, for $n=5$:

$$ \left[ \begin {array}{ccccc} 24&50&35&10&1\\ 48&52& 18&2&0\\ 80&36&4&0&0\\ 72&42&6&0&0 \\ 108&12&0&0&0\\ 96&24&0&0&0 \\ 120&0&0&0&0\end {array} \right] $$

Columns are labelled by number of orbits, $k$, increasing; rows are labelled by cycle type $\lambda$, starting from $1^n$. So a permutation with cycletype $(2,2,1)$ has 80 transitive factorizations and 36 factorizations with 2 orbits.