Combinatorial methods of character enumeration for the unitriangular group

Let ${\text{UT}}_n(q)$ denote the group of unipotent $n\times n$ upper triangular matrices over a field with $q$ elements. The degrees of the complex irreducible characters of ${\text{UT}}_n(q)$ 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 been conjectured that the number of irreducible characters of ${\text{UT}}_n(q)$ with degree $q^e$ is a polynomial in $q-1$ with nonnegative integer coefficients (depending on $n$ and $e$). We confirm this conjecture when $e\le 8$ and $n$ is arbitrary by a computer calculation. In particular, we describe an algorithm which allows us to derive explicit bivariate polynomials in $n$ and $q$ giving the number of irreducible characters of ${\text{UT}}_n(q)$ with degree $q^e$ when $n>2e$ and $e\leq 8$\mathrm{.}Whendividedby$q^{n-e-2}$andwrittenintermsofthevariables$n-2e-1$and$q-1$, these functions are actually bivariate polynomials with nonnegative integer coefficients, suggesting an even stronger conjecture concerning such character counts. As an application of these calculations, we are able to show that all irreducible characters of ${\text{UT}}_n(q)$ with degree $\le q^8$ are Kirillov functions. We also discuss some related results concerning the problem of counting the irreducible constituents of individual supercharacters of ${\text{UT}}_n(q)$.