The number of congruence classes in \(M_ n(\mathbb{F}_ q)\)
The author calls two matrices \(A\) and \(B \in M_ n (K)\) congruent, when \(B = PAP^ t\) for some \(P \in GL_ n (K)\). The number of congruence classes is equal to the number of bilinear functionals on \(K^ n\). It is shown, that in the case of \(K = \mathbb{F}_ q\), this number is the coefficient of \(t^ n\) in \(\prod_{k \geq 1} (1 + t^ k)^ e (1 - qt^{2k}) (1 - t^ k)^{-1}\), where \(e = 1\) for \(q\) even and \(e = 2\) for \(q\) odd. For \(q\) even the number of classes where \(B^{-1} B^ t\) is unipotent is equal to the number of partitions of \(n\).
scientific article; zbMATH DE number 720502