## 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.

Original language | English |
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Pages (from-to) | 247-264 |

Number of pages | 18 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 116 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2009 |

### Bibliographical note

Funding 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.

### Funding

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.

Funders | Funder number |
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National Science Foundation (NSF) | DMS-0604423 |

Directorate for Mathematical and Physical Sciences | 0604423 |

National Security Agency | H98230-06-1-0072 |

## Keywords

- Cyclotomic polynomials
- Descent set statistics
- Fermat primes
- Kummer's theorem
- Multivariate cd-index
- Permutations
- Quasisymmetric functions
- Signed permutations
- Type B quasisymmetric functions

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics