I am looking a formula for $\chi(\tau^{-1}g)$, where $g\in S_n$ and $\chi(g)$ denotes the number of cycles in the cycle notation and let $\tau$ be the cyclic permutation, ie.: $(12\dots n)$. If I have the value of $\chi(g)$, is there any closed expression for $\chi(\tau ^{-1}g)$?
For example, let $n=4$, when $\chi(g)=4$, ie.: $g=id$ then $\chi(\tau^{-1}g)=1$ so I have a pair $(4,1)$, when $\chi(g)=3$ then $\chi(\tau^{-1}g)=2$, thus I have $(3,2)$ pair, etc.
The value of $\chi(\tau g)$ depends on $g$, not just on $\chi(g)$. To see that, see what happens when $g=\tau$ and when $g = \tau^{-1}$.