Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
For $j \in \mathbb{N}$, we denote $\Lambda_j = \left\{i \in \llbracket 1, k \rrbracket \mid \alpha_i \geqslant j\right\}$. Prove that $\operatorname{rg}\left(N_\sigma^j\right) = \sum_{i \in \Lambda_j} \left(\alpha_i - j\right)$.