Euler-Mahonian statistics via polyhedral geometry

Matthias Beck; Benjamin Braun

doi:10.1016/j.aim.2013.06.002

Euler-Mahonian statistics via polyhedral geometry

Matthias Beck, Benjamin Braun

Mathematics

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

10 Scopus citations

Abstract

A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to a pair of such statistics is an Euler-Mahonian distribution, a bivariate polynomial encoding the statistics; such distributions often appear in rational bivariate generating-function identities. We use techniques from polyhedral geometry to establish new multivariate identities generalizing those giving rise to many of the known Euler-Mahonian distributions. The original bivariate identities are then specializations of these multivariate identities. As a consequence of these new techniques we obtain bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics.

Original language	English
Pages (from-to)	925-954
Number of pages	30
Journal	Advances in Mathematics
Volume	244
DOIs	https://doi.org/10.1016/j.aim.2013.06.002
State	Published - 2013

Bibliographical note

Funding Information:
The authors would like to thank Ira Gessel, Carla Savage, and the anonymous referees for their valuable comments and suggestions. This research was partially supported by the NSF through grants DMS-0810105 , DMS-1162638 (Beck) and DMS-0758321 (Braun), and by a SQuaRE at the American Institute of Mathematics.

Funding

The authors would like to thank Ira Gessel, Carla Savage, and the anonymous referees for their valuable comments and suggestions. This research was partially supported by the NSF through grants DMS-0810105 , DMS-1162638 (Beck) and DMS-0758321 (Braun), and by a SQuaRE at the American Institute of Mathematics.

Funders	Funder number
American Institute of Mathematics Structured Quartet Research Ensembles
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	DMS-0810105, DMS-0758321, DMS-1162638

Keywords

Descent statistics
Euler-Mahonian distribution
Eulerian polynomial
Generating functions
Lattice points
Polyhedral geometry

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2013.06.002

Cite this

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title = "Euler-Mahonian statistics via polyhedral geometry",

abstract = "A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to a pair of such statistics is an Euler-Mahonian distribution, a bivariate polynomial encoding the statistics; such distributions often appear in rational bivariate generating-function identities. We use techniques from polyhedral geometry to establish new multivariate identities generalizing those giving rise to many of the known Euler-Mahonian distributions. The original bivariate identities are then specializations of these multivariate identities. As a consequence of these new techniques we obtain bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics.",

keywords = "Descent statistics, Euler-Mahonian distribution, Eulerian polynomial, Generating functions, Lattice points, Polyhedral geometry",

author = "Matthias Beck and Benjamin Braun",

note = "Funding Information: The authors would like to thank Ira Gessel, Carla Savage, and the anonymous referees for their valuable comments and suggestions. This research was partially supported by the NSF through grants DMS-0810105 , DMS-1162638 (Beck) and DMS-0758321 (Braun), and by a SQuaRE at the American Institute of Mathematics.",

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AU - Beck, Matthias

AU - Braun, Benjamin

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N2 - A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to a pair of such statistics is an Euler-Mahonian distribution, a bivariate polynomial encoding the statistics; such distributions often appear in rational bivariate generating-function identities. We use techniques from polyhedral geometry to establish new multivariate identities generalizing those giving rise to many of the known Euler-Mahonian distributions. The original bivariate identities are then specializations of these multivariate identities. As a consequence of these new techniques we obtain bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics.

AB - A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to a pair of such statistics is an Euler-Mahonian distribution, a bivariate polynomial encoding the statistics; such distributions often appear in rational bivariate generating-function identities. We use techniques from polyhedral geometry to establish new multivariate identities generalizing those giving rise to many of the known Euler-Mahonian distributions. The original bivariate identities are then specializations of these multivariate identities. As a consequence of these new techniques we obtain bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics.

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KW - Euler-Mahonian distribution

KW - Eulerian polynomial

KW - Generating functions

KW - Lattice points

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Euler-Mahonian statistics via polyhedral geometry

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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