Cyclotomic factors of the descent set polynomial

TL;DR

This paper studies the factorization patterns of descent set polynomials, revealing connections between cyclotomic factors and binary representations of permutation statistics, with implications for symmetric and signed descent sets.

Contribution

It introduces descent set polynomials as a new encoding method and analyzes their cyclotomic factorization properties, linking them to binary expansions of integers.

Findings

Cyclotomic factors divide descent set polynomials under specific conditions.

The proportion of odd entries in descent set statistics depends only on the binary expansion of n.

Similar properties are observed for signed descent set statistics.

Abstract

We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. As an instance we deduce that the proportion of odd entries in the descent set statistics in the symmetric group S_n only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics.