Circular permutations and irreducible permutation matrices

Question

On a book I'm studying I found this problem: say when a permutation matrix is irreducible. According to the solution provided a permutation matrix is irreducible if and only if it is associated to a circular permutation, so that there are $(n-1)!$ irreducible permutation matrices of order n. But how can we say that a permutation is circular? If I'm not wrong, since they are equivalence classes, you can only count circuar permutations, bu you can't say if a permutation is circular. And if the solution refers to one representative of each equivalence class, which one she is refering to?