I'm working on the problem:
Suppose $V$ is a $123$ dimensional vector space:
$i)$ How many linear maps $T:V \rightarrow V $ are diagonalizable and have $T^2 =0$?
$ii)$ How many linear maps $T:V \rightarrow V $ are not diagonalizable and have $T^2 =0$?
First of all, I reasoned that for part $i)$ we know that there are no diagonalizable nilpotents, hence the answer to part $i)$ is $0$. As for part $ii)$, I know that $T$ must have a Jordan normal form (I don't know to how type up matrices on this) that has Jordan blocks with 1's on the superdiagonal and everything else zero.How can I count how many such arrangements there are?