Find the Jordan normal form of a nilpotent matrix $N$ given the dimensions of the kernels of $N, N^2, N^3$

Since $N$ is nilpotent, it has $0$ as unique eigenvalue. Since $\dim\ker N=3$, by the rank nullity theorem, the rank of $N$ is $7$. Hence the JNF of $N$ has seven 1's. Assume that there are $k$ blocks, and denote by $n_i$ the size of the $i$-th block. Of course, $n_1+\cdots+n_k=10$. Now, the rank of $N$ is $(n_1-1)+(n_2-1)+\cdots+(n_k-1)=10-k$, and we know that this rank is 7 (by rank nullity theorem). Hence k=3. You already got to that point.

Now, let $p$ be the number of blocks of size $1$. Then, the rank of $N^2$ is $(n_1-2)+\cdots+(n_k-2)+p=10-2k+p$. Since $\dim\ker N^2=6$, we can conclude (by the rank–nullity theorem again) that the rank of $N^2$ is $4$. Hence (since $k=3$) we conclude that $p=0$. So all blocks have a size at least 2.

Finally, let $q$ be the number of blocks of size $2$. Then, the rank of $N^3$ is $(n_1-3)+\cdots+(n_k-3)+q=10-3k+q$. Since $\dim\ker N^3=7$, we conclude that the rank of $N^3$ is $3$. Since $k=3$, we conclude that $q=2$. Hence $N$ has 2 blocks of size $2$ and one block of size $6$: we hence conclude that the JNF is (up to reordering of the blocks): $$\left(\begin{array}{cc|cc|cccccc}0&1&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0\\\hline0&0&0&1&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0\\\hline0&0&0&0&0&1&0&0&0&0\\0&0&0&0&0&0&1&0&0&0\\0&0&0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&0&0&1\\0&0&0&0&0&0&0&0&0&0\end{array}\right).$$

This can be seen in term of partitions.

An $n \times n$ nilpotent matrix $N$ can be described via a partition $$ p = (n_{1}, n_{2}, \dots, n_{k}) $$ of $n$, with $n_{1} \ge n_{2} \ge \dots \ge n_{k} > 0$, which records the size of the nilpotent Jordan block in a Jordan normal form.

Now one can show (it is really straightforward) that for the dual partition $q$ of $p$ one has $$ q = (\dim(\ker(N)), \dim(\ker(N^{2})) - \dim(\ker(N)), \dim(\ker(N^{3})) - \dim(\ker(N^{2})), \dots). $$

In your case $$ q = (3, 6-3, 7-6, \dots) = (3, 3, 1, \dots), $$ and thus $$ q = (3, 3, 1, 1, 1, 1). $$ The dual partition is $$ p = (6, 2, 2). $$

Hint Since the matrix $N$ is nilpotent, the only eigenvalue is zero, and as the question points out, the number of blocks is equal to the geometric multiplicity of the eigenvalue zero, which by definition is the dimension of the kernel (in this case, $3$). Denote the sizes of the three Jordan blocks by $k_1, k_2, k_3$.

Computing gives that the Jordan block $J_k$ of eigenvalue $0$ and size $k$ satisfies $\dim \ker (J_1^2) = 1$ and $\dim \ker (J_k^2) = 2$ for $k > 1$. So, the Jordan normal form $J = J_1 \oplus J_2 \oplus J_3$ of $N$ satisfies $$6 = \dim \ker (N^2) = \dim \ker (J^2) = \dim \ker (J_{k_1}^2) + \dim \ker (J_{k_2}^2) + \dim \ker (J_{k_3}^2),$$ and hence we must have $\dim \ker (J_{k_a}^2) = 2$, and therefore $k_a \geq 2$, for all $a$.

We can similarly analyze the kernels of $J_k^3$, which in this case turns out to be enough to determine the Jordan normal form $J$.