© 2015

Eulerian Numbers


  • Offers a self-contained treatment of progress and problems related to the Eulerian numbers

  • Covers a topic that plays an important role in combinatorics, number theory, and topology

  • Provides previously-unpublished coverage of gamma-nonnegativity of a simplicial complex and its results in combinatorial terms

  • Includes discussion of open problems and directions for future research, as well as numerous exercises and examples


Part of the Birkhäuser Advanced Texts Basler Lehrbücher book series (BAT)

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


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.​


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

Authors and affiliations

  1. 1.Department of Mathematical SciencesDePaul UniversityChicagoUSA

About the authors

T. Kyle Petersen is an Associate Professor of Mathematics at DePaul University, Chicago, USA.  His research areas include algebraic, enumerative, and topological combinatorics.  He received his PhD in Mathematics from Brandeis University.

Bibliographic information


“This book is a beautiful modern collection about Eulerian numbers, which reflects its author's interest and expertise in the subject. The material is rich. Eulerian numbers and their close relatives, such as Narayana numbers, show up in a variety of topics from enumerative, algebraic, topological, and geometric combinatorics. … It is self-contained and presented in a clear ... way. The author describes everything from the basics, so the text should be accessible to a broad audience interested in combinatorics.” (Damir Yeliussizov, Mathematical Reviews, April, 2017)

“This work offers wonderful material to supplement a course in combinatorics or discrete mathematics, or for a stimulating capstone course. Summing Up: Highly recommended. Lower-division undergraduates through researchers/faculty.” (D. V. Feldman, Choice, Vol. 53 (11), July, 2016)

“This book serves dual purposes. On the one hand, it is a monograph on Eulerian numbers and their generalizations. … On the other hand, the book gives an introduction to contemporary enumerative, algebraic and geometric combinatorics, and it can be used as a text at beginning graduate or advanced undergraduate level. … This is a well-written text for a good course.” (László A. Székely, zbMATH 1337.05001, 2016)