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Let $\UT_n(q)$ denote the group of unipotent $n\times n$ upper triangularmatrices over a field with $q$ elements. The degrees of the complex irreduciblecharacters of $\UT_n(q)$ are precisely the integers $q^e$ with $0\leq e\leq\lfloor \frac{n}{2} \rfloor \lfloor \frac{n-1}{2} \rfloor$, and it has beenconjectured that the number of irreducible characters of $\UT_n(q)$ with degree$q^e$ is a polynomial in $q-1$ with nonnegative integer coefficients (dependingon $n$ and $e$). We confirm this conjecture when $e\leq 8$ and $n$ is arbitraryby a computer calculation. In particular, we describe an algorithm which allowsus to derive explicit bivariate polynomials in $n$ and $q$ giving the number ofirreducible characters of $\UT_n(q)$ with degree $q^e$ when $n>2e$ and $e\leq8$. When divided by $q^{n-e-2}$ and written in terms of the variables $n-2e-1$and $q-1$, these functions are actually bivariate polynomials with nonnegativeinteger coefficients, suggesting an even stronger conjecture concerning suchcharacter counts. As an application of these calculations, we are able to showthat all irreducible characters of $\UT_n(q)$ with degree $\leq q^8$ areKirillov functions. We also discuss some related results concerning the problemof counting the irreducible constituents of individual supercharacters of$\UT_n(q)$.

Combinatorial methods of character enumeration for the unitriangular group