The Number of Congruence Classes in Mn( q)

Call matrices A and B congruent when PAPt = B for some invertible P. Extending a result of Gow, this paper shows that the number of congruence classes in the n × n matrices over q is the coefficient of tn in [formula] (where e = 1 for even q and e = 2 for odd q. For even q, the number of classes where B−1Bt is unipotent is equal to the number of partitions of n.