On the proportion of derangements in affine classical groups

We derive exact formulas for the proportions of derangements and of derangements of $p$-power order in the affine classical groups $A{U}_m(q)$, $AS{p}_{2m}(q)$, $A{O}_{2m+1}(q)$ and $A{O}_{2m}^{\pm }(q)$, where $p$ denotes the characteristic of the defining finite field. In the unitary case, the formulas rely on a result on partitions of independent interest: we obtain a generating function for integer partitions $\lambda =({\lambda }_1,\dots ,{\lambda }_m)$ into $m$ parts, with ${\lambda }_1\ge \cdots \ge {\lambda }_m$, such that either ${\lambda }_1=1$ or ${\lambda }_{k-1}>{\lambda }_k=k$ for some $k\in \{2,\dots ,m\}$. In the symplectic and orthogonal cases, the proofs of the formulas reduce to verifying three $q$-polynomial identities conjectured by the author and later proved by Fulman and Stanton.