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 $n$. Prove that, for every $j \in \mathbb{N}^*$, the integer $d_j = \operatorname{rg}\left(u^{j-1}\right) - \operatorname{rg}\left(u^j\right)$ is equal to the number of blocks $J_{\alpha_i}$ whose size $\alpha_i$ is greater than or equal to $j$.
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 $n$.
Prove that, for every $j \in \mathbb{N}^*$, the integer $d_j = \operatorname{rg}\left(u^{j-1}\right) - \operatorname{rg}\left(u^j\right)$ is equal to the number of blocks $J_{\alpha_i}$ whose size $\alpha_i$ is greater than or equal to $j$.