Cyclotomic factors of the descent set polynomial

Denis Chebikin; Richard Ehrenborg; Pavlo Pylyavskyy; Margaret Readdy

doi:10.1016/j.jcta.2008.05.011

Cyclotomic factors of the descent set polynomial

Denis Chebikin, Richard Ehrenborg, Pavlo Pylyavskyy, Margaret Readdy

School of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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 (US)
Pages (from-to)	247-264
Number of pages	18
Journal	Journal of Combinatorial Theory. Series A
Volume	116
Issue number	2
DOIs	https://doi.org/10.1016/j.jcta.2008.05.011
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.

Keywords

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

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10.1016/j.jcta.2008.05.011

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title = "Cyclotomic factors of the descent set polynomial",

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 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.",

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

author = "Denis Chebikin and Richard Ehrenborg and Pavlo Pylyavskyy and Margaret Readdy",

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.",

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doi = "10.1016/j.jcta.2008.05.011",

language = "English (US)",

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journal = "Journal of Combinatorial Theory. Series A",

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AU - Chebikin, Denis

AU - Ehrenborg, Richard

AU - Pylyavskyy, Pavlo

AU - Readdy, Margaret

N1 - 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.

PY - 2009/2

Y1 - 2009/2

N2 - 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.

AB - 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.

KW - Cyclotomic polynomials

KW - Descent set statistics

KW - Fermat primes

KW - Kummer's theorem

KW - Multivariate cd-index

KW - Permutations

KW - Quasisymmetric functions

KW - Signed permutations

KW - Type B quasisymmetric functions

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JO - Journal of Combinatorial Theory. Series A

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ER -

Cyclotomic factors of the descent set polynomial

Abstract

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