## Cycle index series

Prove that for the group $C_n = \left<(123 \dots n)\right>$
The Cycle Index Series is given by $Z_{C_n}(t_1 \dots t_n) = \frac{1}{n}\sum_{d|n}\phi(d){t_d}^{\frac{n}{d}}$

Basically this comes down to showing if $c \in C_n$ and $|c| = k$, then when c is written as a product of disjoint cycles, there are $\frac{n}{k}$ cycles of length $k$
I.e, $c = \epsilon_1 \dots \epsilon_{\frac{n}{k}}$
But I can't seem to show this.

The part with the Euler phi function is easy to show, I don't need help on that.