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 ${\mathrm{\Phi }}_{2s}$ or ${\mathrm{\Phi }}_{4s}$ where $s$ is an odd integer, with many of these being of the form ${\mathrm{\Phi }}_{2p}$ where $p$ is a prime. We also show that if ${\mathrm{\Phi }}_2$ is a factor of $Q_{2n}(t)$ then it is a double factor. Finally, we give conditions for an odd prime power $q = p^{r}$forwhich$\Phi_{2p}$isadoublefactorof$Q_{2q}(t)$andof$Q_{q+1}(t)$.