On the proportion of derangements in affine classical groups

Abstract We derive exact formulas for the proportions of derangements and of derangements of p -power order in the affine classical groups $\operatorname {\mathrm {AU}}_m(q)$ , $\operatorname {\mathrm {ASp}}_{2m}(q)$ , $\operatorname {\mathrm {AO}}_{2m+1}(q)$ and $\operatorname {\mathrm {AO}}^{\pm }_{2m}(q)$ , where p denotes the characteristic of the defining finite field. In the unitary case, the proofs of 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 $\lambda _1\ge \dots \ge \lambda _m$ , 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.