= Solution
The <generating function> for <integer partitions> is $\sum_\lambda u^{|\lambda|}=\prod_{i\ge1}(1-u^i)^{-1}$, since independently choosing the multiplicity of each part size gives a <geometric series>. Applying the parameterization in 5(a),
$$
\sum_{n\ge0}k_n(q)t^n=\prod_{f\ne t}\prod_{i\ge1}(1-t^{i\deg(f)})^{-1}.
$$
Unique factorization of <monic polynomials> shows
$$
\prod_{f\ne t}(1-u^{\deg(f)})^{-1}
=1+\sum_{r\ge1}(q-1)q^{r-1}u^r
=\frac{1-u}{1-qu}.
$$
Indeed the left side counts <monic polynomials> with nonzero constant term; in degree $r\ge1$ there are $(q-1)q^{r-1}$ choices. Reordering the formal products is legitimate because only finitely many factors contribute to any fixed coefficient. Substituting $u=t^i$ for each $i$ gives the <conjugacy-class generating function for finite general linear groups>:
$$
\boxed{k_n(q)=[t^n]\prod_{i\ge1}\frac{1-t^i}{1-qt^i}.}
$$
The constant term corresponds to the trivial zero-dimensional <group>.
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