On the proportion of derangements in affine classical groups

Abstract: We derive explicit formulas for the proportion of derangements and of derangements of $p$-power order in the affine unitary group $AU_m(q)$ over the finite field $\mathbb{F}{q^2}$ of characteristic $p$. Both formulas rely on a result of independent interest concerning integer partitions: we obtain a generating function for partitions $\lambda=(\lambda_1, \dots, \lambda_m)$ into $m$ parts, for which $\lambda_1=1$ or there exists $k \in {2, \dots,m}$ such that $\lambda{k-1}>\lambda_k=k$. Moreover, when $p$ is odd, we conjecture remarkably simple formulas for the proportions of derangements in the finite affine symplectic group $ASp_{2m}(q)$, and in the affine orthogonal groups $AO_{2m+1}(q)$ and $AO^{\pm}_{2m}(q)$, and we also propose conjectures for the proportions of derangements of $p$-power order. We further reduce the proofs of these conjectures to establishing three $q$-polynomial identities.