Conjugacy classes of $S_n$ under the action of $S_{n-1}$

Question

I try to get explicitly сonjugacy classes of $S_n$ under the action of $S_{n-1}$. I believe that in the description of the classes present cycle type of a permutation and yet another parameter. But I can not get the final result.

Answer 1

What conjugation by an element of $S_{n-1}$ preserves is the cycle structure plus the length of the cycle that position $n$ is part of.

Thus each conjugacy class of $S_n$ splits into one class for each different length cycle in its cycle structure.