  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.
