For a given $n$, is there any characterization for the commutative subalgebras of $M_n(\Bbb{C})$? I would like to know how many commutative subalgebras there are for each possible dimension.

In view of Chapman's answer, I am refining my previous question:

Given $k\leq n$, is there any way of describing the commutative subalgebras of $M_n$ which are of dimension $k$.