Why is $\mathcal{P}$ = SET($\mathcal{Z}$)?

Question

in the book Analytic Combinatorics, where the symbolic method is described, they say that $\mathcal{P}$ = $SET(\mathcal{Z})$ and therefore $\hat{P}(z) = exp(\hat{Z}(z))$. $\mathcal{P}$ denotes the family of permutations of finite sets [n] and $\mathcal{Z}$ the cyclic permutations of [n]. So my question is, why this is the case? How can you think about it in order to come to that conclusion?