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 Sn only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics.
Bibliographical noteFunding Information:
The authors thank the referee for improving the proof of Theorem 2.1. The authors also thank the MIT Mathematics Department where this research was carried out. The second author was partially supported by National Security Agency grant H98230-06-1-0072, and the third author was partially supported by National Science Foundation grant DMS-0604423.
Copyright 2008 Elsevier B.V., All rights reserved.
- Cyclotomic polynomials
- Descent set statistics
- Fermat primes
- Kummer's theorem
- Multivariate cd-index
- Quasisymmetric functions
- Signed permutations
- Type B quasisymmetric functions