  Let $U_n$ denote the group of upper $n \times n$ unitriangular matrices over
a fixed finite field $\mathbb{F}$ of order $q$. That is, $U_n$ consists of
upper triangular $n \times n$ matrices having every diagonal entry equal to
$1$. It is known that the degrees of all irreducible complex characters of
$U_n$ are powers of $q$. It was conjectured by Lehrer that the number of
irreducible characters of $U_n$ of degree $q^e$ is an integer polynomial in $q$
depending only on $e$ and $n$. We show that there exist recursive (for $n$)
formulas that this number satisfies when $e$ is one of $1, 2$ and $3$, and thus
show that the conjecture is true in those cases.
