Suppose a $4$ by $4$ matrix $A$ is nilpotent and upper triangular, and all $(i, j)$ entries for $i < j$ are chosen randomly and uniformly in the interval $[−1, 1]$. What are the probabilities that its Jordan canonical form corresponds to the partitions \begin{gather*} 4 = 4, \qquad 4 = 3 + 1, \qquad 4 = 2 + 2, \\ 4 = 2 + 1 + 1, \qquad 4 = 1 + 1 + 1 + 1? \end{gather*}
I don't have any ideas about this question. Please give me any insights or ideas.