The variety of nilpotent $n\times n$ matrices $\mathcal{N}_n$ over an algebraically closed field $k$ is the disjoint union of orbits under the action of conjugation by $GL_n(k)$. These orbits are characterized by the Jordan form. Since all eigenvalues of a nilpotent matrix are zero, the data of the Jordan form of a nilpotent $n\times n$ matrix is equivalent to a partition of $n$.
If we place a partial order on the orbits of $\mathcal{N}_n$ by $\mathcal{O}\le\mathcal{O}'$ if and only if $\overline{\mathcal{O}}\subset\overline{\mathcal{O}'}$, then it turns out that the partial order on the orbits corresponds to the dominance ordering on partitions.
Now, suppose that $\mathcal{C}_2(\mathcal{N}_n)$ is the variety of commuting ordered pairs of nilpotent $n\times n$ matrices. $GL_n$ still acts by conjugation component-wise, and we can place the same partial order on orbits mentioned above. Namely $\mathcal{O}\le\mathcal{O}'$ if and only if $\overline{\mathcal{O}}\subset\overline{\mathcal{O}'}$. As in the previous case, is there a description of the partial order in terms of partitions? Does anybody know of a reference where this question is considered?