Counting representations of $k[x,y]$ when $k$ is finite $\newcommand{\GFq}{\mathbf F_q}$
Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial $p_n(t)$ (with say, integer coefficients) such that $r_n(q) = p_n(q)$ for all prime powers $q$?
Here are the values that I know (see arXiv:1212.6157v2):


*

*$r_1(q) = q^2$

*$r_2(q) = q^4 + q^3 + q^2$

*$r_3(q) = q^6 + q^5 + 2q^4 + q^3 + 2q^2$

*$r_4(q) = q^8 + q^7 + 3q^6 + 3q^5 + 5q^4 + 3q^3 + 3q^2$


Note: for the free algebra $\GFq\langle x, y\rangle$ this is well known, since it is the number of $n$-dimensional representations of the quiver with one vertex and two loops. In fact, it is also known now that this polynomial has non-negative integer coefficients (this follows from the work of Mozgovoy or Hausel, Letellier and Rodriguez-Villegas).
 A: The number you seek is closely related to the number of commuting pairs of $n\times n$ matrices over $\mathbf{F}_q$. This is a polynomial and the formula was first given by Feit and Fine in a 1960 Duke paper. The stack of $\mathbf{F}_q[x,y]$ modules is the stack quotient of the commuting variety 
$$C(n) = \{(A,B)\in End(\mathbf{F}_q^n)^2 : [A,B]=0\}$$
by $Gl(n)$ acting by conjugation. The number you have asked for is the number of points in the coarse space of this stack. I don't know much about that number, but the "stacky" number of points is much better behaved --- in other words, instead of counting modules, you should be counting modules by 1 over the number of automorphisms of the module. That number is equal to the number of commuting pairs of matrices over $\mathbf{F}_q$ divided by the number of elements in $Gl(n,\mathbf{F}_q)$. The generating function of that number has a nice product expansion:
$$\sum_{n=0}^{\infty} \frac{|C(n)|}{|Gl(n)|} t^n = \prod_{k=1}^\infty \prod_{m=1}^\infty (1-q^{2-m}t^k)^{-1}$$
This is in Feit and Fine's paper or you can find a more modern motivic derivation of this in my paper with Morrison. http://arxiv.org/abs/1206.5864
