Counting characters of upper triangular groups

Working over a finite field of order q, let Un(q) be the group of upper triangular n×n matrices with all diagonal entries equal to 1. It is known that all complex irreducible characters of Un(q) have degrees that are powers of q and that all powers qe occur for 0⩽e⩽μ(n), where the upper bound μ(n) is defined by the formulas μ(2m)=m(m−1) and μ(2m+1)=m2. It has been conjectured that the number of irreducible characters having degree qe is some polynomial in q with integer coefficients (depending on n and e). In this paper, we construct explicit polynomials for the cases where e is one of μ(n), μ(n)−1 and 1. Also, a list of polynomials is presented that appear to give the correct character counts for n⩽9. Several related results concerning pattern groups are also included.