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Let $q$ be a power of a prime $p$. Let $P$ be a parabolic subgroup of thegeneral linear group $\GL_n(q)$ that is the stabilizer of a flag in $\FF_q^n$of length at most 5, and let $U = O_p(P)$. In this note we prove that, as afunction of $q$, the number $k(U)$ of conjugacy classes of $U$ is a polynomialin $q$ with integer coefficients.

Counting conjugacy classes in the unipotent radical of parabolic subgroups of $\GL_n(q)$