For $\sigma \in \mathfrak{S}_n$ and $\ell \in \llbracket 2, n \rrbracket$, we denote $c_\ell(\sigma)$ the number of cycles of length $\ell$ in the decomposition of $\sigma$ into cycles with disjoint supports. We denote $c_1(\sigma)$ the number of fixed points of $\sigma$: $$c_1(\sigma) = \operatorname{Card}\{j \in \llbracket 1, n \rrbracket, \sigma(j) = j\}.$$ Show that $\sigma \in \mathfrak{S}_n$ and $\tau \in \mathfrak{S}_n$ are conjugate if and only if, for all $\ell \in \llbracket 1, n \rrbracket, c_\ell(\sigma) = c_\ell(\tau)$.
For $\sigma \in \mathfrak{S}_n$ and $\ell \in \llbracket 2, n \rrbracket$, we denote $c_\ell(\sigma)$ the number of cycles of length $\ell$ in the decomposition of $\sigma$ into cycles with disjoint supports. We denote $c_1(\sigma)$ the number of fixed points of $\sigma$:
$$c_1(\sigma) = \operatorname{Card}\{j \in \llbracket 1, n \rrbracket, \sigma(j) = j\}.$$
Show that $\sigma \in \mathfrak{S}_n$ and $\tau \in \mathfrak{S}_n$ are conjugate if and only if, for all $\ell \in \llbracket 1, n \rrbracket, c_\ell(\sigma) = c_\ell(\tau)$.