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.