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

Mathematics

Producción científica: Article › revisión exhaustiva

3 Citas (Scopus)

Resumen

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.

Idioma original	English
Páginas (desde-hasta)	247-264
Número de páginas	18
Publicación	Journal of Combinatorial Theory. Series A
Volumen	116
N.º	2
DOI	https://doi.org/10.1016/j.jcta.2008.05.011
Estado	Published - feb 2009

Nota bibliográfica

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.

Financiación

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.

Financiadores	Número del financiador
National Science Foundation Arctic Social Science Program	DMS-0604423
National Security Agency	H98230-06-1-0072

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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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",

volume = "116",

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T1 - Cyclotomic factors of the descent set polynomial

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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DO - 10.1016/j.jcta.2008.05.011

M3 - Article

AN - SCOPUS:56549108450

SN - 0097-3165

VL - 116

SP - 247

EP - 264

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

IS - 2

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Cyclotomic factors of the descent set polynomial

Resumen

Nota bibliográfica

Financiación

ASJC Scopus subject areas

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