Prove that for the group $\displaystyle C_n = \left<(123 \dots n)\right>$

The Cycle Index Series is given by $\displaystyle Z_{C_n}(t_1 \dots t_n) = \frac{1}{n}\sum_{d|n}\phi(d){t_d}^{\frac{n}{d}}$

Basically this comes down to showing if $\displaystyle c \in C_n$ and $\displaystyle |c| = k$, then when c is written as a product of disjoint cycles, there are $\displaystyle \frac{n}{k} $ cycles of length $\displaystyle k$

I.e, $\displaystyle c = \epsilon_1 \dots \epsilon_{\frac{n}{k}}$

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.