Let $p=k_1+\dotsb+k_r$ and $m$ an integer. Consider the big rotation $\sigma:=(1\,\dotsb\,p)$. I want to find another permutation $\tau$ with the following two properties:
- $\tau$ has exactly $r$ cycles with lengths $k_1,\dotsc,k_r$,
- $\tau\circ\sigma$ has exactly $m$ cycles (counting trivial cycles).
Property (1) determines $\tau$ uniquely up to conjugation, but maybe there is still a chance to find the right $\tau$ …