Table of contents

  1. Front Matter
    Pages i-xviii
  2. Combinatorics

    1. Front Matter
      Pages 1-1
    2. T. Kyle Petersen
      Pages 3-18
    3. T. Kyle Petersen
      Pages 19-45
    4. T. Kyle Petersen
      Pages 47-69
    5. T. Kyle Petersen
      Pages 71-93
    6. T. Kyle Petersen
      Pages 127-149
  3. Combinatorial topology

    1. Front Matter
      Pages 161-161
    2. T. Kyle Petersen
      Pages 163-183
    3. T. Kyle Petersen
      Pages 185-202
    4. T. Kyle Petersen
      Pages 203-233
  4. Coxeter groups

    1. Front Matter
      Pages 235-235
    2. T. Kyle Petersen
      Pages 237-272
    3. T. Kyle Petersen
      Pages 273-291
    4. T. Kyle Petersen
      Pages 293-331
  5. Back Matter
    Pages 347-456

About this book

Introduction

This text presents the Eulerian numbers in the context of modern enumerative, algebraic, and geometric combinatorics. The book first studies Eulerian numbers from a purely combinatorial point of view, then embarks on a tour of how these numbers arise in the study of hyperplane arrangements, polytopes, and simplicial complexes. Some topics include a thorough discussion of gamma-nonnegativity and real-rootedness for Eulerian polynomials, as well as the weak order and the shard intersection order of the symmetric group.

The book also includes a parallel story of Catalan combinatorics, wherein the Eulerian numbers are replaced with Narayana numbers. Again there is a progression from combinatorics to geometry, including discussion of the associahedron and the lattice of noncrossing partitions.

The final chapters discuss how both the Eulerian and Narayana numbers have analogues in any finite Coxeter group, with many of the same enumerative and geometric properties. There are four supplemental chapters throughout, which survey more advanced topics, including some open problems in combinatorial topology.

This textbook will serve a resource for experts in the field as well as for graduate students and others hoping to learn about these topics for the first time.​

Keywords

Catalan numbers Coxeter groups Eulerian numbers Gal's conjecture Gessel's conjecture Narayana numbers Simplicial complex gamma-nonnegativity

Authors and affiliations

  • T. Kyle Petersen
    • 1
  1. 1.Department of Mathematical SciencesDePaul UniversityChicagoUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4939-3091-3
  • Copyright Information Springer Science+Business Media New York 2015
  • Publisher Name Birkhäuser, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4939-3090-6
  • Online ISBN 978-1-4939-3091-3
  • Series Print ISSN 1019-6242
  • Series Online ISSN 2296-4894
  • About this book