The descent set polynomial revisited

TL;DR

This paper investigates cyclotomic factors in the descent set polynomial, identifying classes of factors and conditions for their multiplicity, thereby advancing understanding of its algebraic structure.

Contribution

It provides new classes of cyclotomic factors in the descent set polynomial and establishes conditions for their multiplicity and specific factors.

Findings

Identifies large classes of factors of the form Φ_{2s} or Φ_{4s} with s odd.

Shows that if Φ_2 divides Q_{2n}(t), it does so with multiplicity two.

Provides conditions for Φ_{2p} to be a double factor of Q_{2q}(t) and Q_{q+1}(t).

Abstract

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 $Q_{q+1}(t)$.