Conjugacy classes of the symmetric group correspond to cycle types

Statement

Let $\sigma, \tau \in S_n$. Then $\sigma$ and $\tau$ are conjugate in $S_n$ if and only if they have the same cycle type.

Remarks

The key formula is $\pi (a_1\ a_2\ \dots\ a_k) \pi^{-1} = (\pi(a_1)\ \pi(a_2)\ \dots\ \pi(a_k))$ for each $\pi \in S_n$. So conjugation keeps the cycle type, and a suitable $\pi$ sends the cycles of $\sigma$ onto the cycles of $\tau$. Thus the number of conjugacy classes of $S_n$ is the number of partitions of $n$. For example, $S_4$ has $5$ conjugacy classes.

Explanations

L1 human

Conjugation by $\pi$ renames the points, but it keeps the shape of each cycle. Conversely, two permutations of the same shape differ only by a renaming of the points. So in $S_n$, conjugate means "same shape".