The descent set polynomial revisited

We continue to explore cyclotomic factors in the descent set polynomial $Q_{n}(t)$, which was introduced by Chebikin, Ehrenborg, Pylyavskyy and Readdy. We obtain large classes of factors of the form $\Phi_{2s}$ or $\Phi_{4s}$ where $s$ is an odd integer, with many of these being of the form $\Phi_{2p}$ where $p$ is a prime. We also show that if $\Phi_{2}$ is a factor of $Q_{2n}(t)$ then it is a double factor. Finally, we give conditions for an odd prime power $q = p^{r}$ for which $\Phi_{2p}$ is a double factor of $Q_{2q}(t)$ and of $Q_{q+1}(t)$.