Conjugacy-class generating function for finite general linear groups

If $k_n(q)$ counts conjugacy classes in $G{L}_n(q)$, then ${\sum }_{n\ge 0}{k}_n(q)t^n={\prod }_{i\ge 1}(1-t^i)/(1-q{t}^i)$. Associate a partition to each irreducible polynomial other than t. The Euler product over those polynomials counts monic polynomials with nonzero constant term and is $(1-u)/(1-qu)$.