I've been doing some work counting commuting pairs of unitriangular matrices over $GL_{n}(F_{p})$. So far, I believe that for $n=2$, there are $p^2$ such pairs, and for $n=3$ there are $p^5+p^4-p^3$ such pairs. Can anybody recognize these polynomials, generalize to arbitrary $n<p$?
Recall that unitriangular matrices are upper-triangular matrices of having entries of $1$ on the diagonal.
Any help would be appreciated so much. Thank you all.