Kernel dimensions of powers of a nilpotent map ($d_i=\dim \ker T^i$)

For a nilpotent linear map with minimal polynomial $x^m$, set $K_i=\ker T^i$ and $d_i=\dim K_i$, with $d_0=0$. ThenFor strictness, choose $w$ with $T^{m-1}w\ne 0$. The vector $T^{m-1-i}w$ belongs to $K_{i+1}\setminus K_i$ for $0\le i<m$. For the second inequality, the rank-nullity theorem givesThe first image is contained in the second because $T(K_{i+1})\subseteq K_i$.

If the Nilpotent Jordan block sizes are $s_1,\dots ,s_r$, thenThus successive increments count the Jordan blocks reaching a given size, and $(d_i-d_{i-1})-(d_{i+1}-d_i)$ counts blocks of size exactly $i$, with $d_{m+1}=d_m$. The nullities determine the entire Jordan normal form. Conversely any positive nonincreasing integer increments summing to $\dim V$ are realized by taking the corresponding numbers of each Nilpotent Jordan block.