Euler-Mahonian statistics via polyhedral geometry

Matthias Beck, Benjamin Braun

Research output: Contribution to journalArticlepeer-review

8 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 languageEnglish
Pages (from-to)925-954
Number of pages30
JournalAdvances in Mathematics
Volume244
DOIs
StatePublished - 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.

Keywords

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

ASJC Scopus subject areas

  • General Mathematics

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