= Solution
The increments $\Delta_i=d_i-d_{i-1}$ are positive integers by part (a)(i), are nonincreasing by part (a)(ii), and sum to $d_m=4$. Thus they form an <integer partition> of $4$. The complete list is $(4)$, $(3,1)$, $(2,2)$, $(2,1,1)$, and $(1,1,1,1)$.
To prove all five occur over every <field>, let $J_s(0)$ be a <Nilpotent Jordan block> of size $s$. It sends each <standard basis> vector to its predecessor, with the first sent to zero, so
$$
\dim\ker J_s(0)^i=\min(i,s).
$$
For a <block diagonal matrix>, the <nullities> add. The following explicit <nilpotent matrices> therefore realize every possible sequence:
$$
\boxed{\begin{array}{c|c|c}
m&(d_1,\ldots,d_m)&\text{realizing matrix}\\\hline
1&(4)&0_{4\times4}\\
2&(3,4)&J_2(0)\oplus J_1(0)\oplus J_1(0)\\
2&(2,4)&J_2(0)\oplus J_2(0)\\
3&(2,3,4)&J_3(0)\oplus J_1(0)\\
4&(1,2,3,4)&J_4(0)
\end{array}}.
$$
In each row the largest <Jordan block> size is exactly the <nilpotency index> $m$, so the displayed sequence stops at the correct place.
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