Handbook of Enumerative Combinatorics: 1st Edition (Hardback) book cover

Handbook of Enumerative Combinatorics

1st Edition

Edited by Miklos Bona

Chapman and Hall/CRC

1,088 pages | 225 B/W Illus.

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Presenting the state of the art, the Handbook of Enumerative Combinatorics brings together the work of today’s most prominent researchers. The contributors survey the methods of combinatorial enumeration along with the most frequent applications of these methods.

This important new work is edited by Miklós Bóna of the University of Florida where he is a member of the Academy of Distinguished Teaching Scholars. He received his Ph.D. in mathematics at Massachusetts Institute of Technology in 1997. Miklós is the author of four books and more than 65 research articles, including the award-winning Combinatorics of Permutations. Miklós Bóna is an editor-in-chief for the Electronic Journal of Combinatorics and Series Editor of the Discrete Mathematics and Its Applications Series for CRC Press/Chapman and Hall.

The first two chapters provide a comprehensive overview of the most frequently used methods in combinatorial enumeration, including algebraic, geometric, and analytic methods. These chapters survey generating functions, methods from linear algebra, partially ordered sets, polytopes, hyperplane arrangements, and matroids. Subsequent chapters illustrate applications of these methods for counting a wide array of objects.

The contributors for this book represent an international spectrum of researchers with strong histories of results. The chapters are organized so readers advance from the more general ones, namely enumeration methods, towards the more specialized ones.

Topics include coverage of asymptotic normality in enumeration, planar maps, graph enumeration, Young tableaux, unimodality, log-concavity, real zeros, asymptotic normality, trees, generalized Catalan paths, computerized enumeration schemes, enumeration of various graph classes, words, tilings, pattern avoidance, computer algebra, and parking functions.

This book will be beneficial to a wide audience. It will appeal to experts on the topic interested in learning more about the finer points, readers interested in a systematic and organized treatment of the topic, and novices who are new to the field.


"Mathematical handbooks are among the most essential library resources, providing compilations of formulas, tables, graphs, etc. Traditional handbooks speak equally to experts and casual users of mathematics. Other handbooks, such as the current work, are really encyclopedic compendiums of survey articles primarily addressing readers who make mathematics their main business. They supplement systematic monographs that develop subjects methodically but require extreme reader commitment and journal literature that provides quick access to specific results for those with prerequisite knowledge. Researchers will benefit from rapid authoritative citations to newer or lesser-known results. Students, undergraduate and graduate, will find accessible, systematic snapshots of whole subjects, helping them discover what they most wish to learn and, equally, what they will then need to learn on the way. Enumerative combinatorics means counting problems, so that subject begins classically with permutations and combinations but is active now with connections to probability, graph theory, statistical mechanics, geometry, representation theory, analysis, and computer science. Chapters here divide between general counting methods, both exact and approximate, and special classes of objects for counting via any suitable means. The volume, part of the 'Discrete Mathematics and Its Applications' series, is well edited by Bóna (Univ. of Florida), who successfully pools the expertise of leaders in the field. Summing up: Recommended. Upper-division undergraduates through professionals/practitioners."

—D. V. Feldman, University of New Hampshire, Durham, USA, for CHOICE, March 2016

"I cannot think of any topic that I would like to have seen presented here that the book omits. The chapters discuss not only methods in the study of enumerative combinatorics, but also objects that lend themselves to study along these lines. … accessible to a wide audience … this will clearly be a book that anybody with a serious interest in combinatorics will want to have on his or her bookshelf, and of course it belongs in any self-respecting university library. Having seen firsthand what it takes to edit a handbook like this, I know that Miklós Bóna must have invested a great deal of time and effort in the creation of this volume, as did the authors of the individual chapters. Their efforts have not been in vain; this is a valuable book."

MAA Reviews, July 2015

Table of Contents


Algebraic and Geometric Methods in Enumerative Combinatorics


What is a Good Answer?

Generating Functions

Linear Algebra Methods



Hyperplane Arrangements



Analytic Methods; Helmut Prodinger


Combinatorial Constructions and Associated Ordinary Generating Functions

Combinatorial Constructions and Associated Exponential Generating Functions

Partitions and Q-Series

Some Applications of the Adding a Slice Technique

Lagrange Inversion Formula

Lattice Path Enumeration: The Continued Fraction Theorem

Lattice Path Enumeration: The Kernel Method

Gamma and Zeta Function

Harmonic Numbers and Their Generating Functions

Approximation of Binomial Coefficients

Mellin Transform and Asymptotics of Harmonic Sums

The Mellin-Perron Formula

Mellin-Perron Formula: Divide-and-Conquer Recursions

Rice’s Method

Approximate Counting

Singularity Analysis of Generating Functions

Longest Runs in Words

Inversions in Permutations and Pumping Moments

Tree Function

The Saddle Point Method

Hwang’s Quasi-Power Theorem


Asymptotic Normality in Enumeration; E. Rodney Canfield

The Normal Distribution

Method 1: Direct Approach

Method 2: Negative Roots

Method 3: Moments

Method 4: Singularity Analysis

Local Limit Theorems

Multivariate Asymptotic Normality

Normality in Service to Approximate Enumeration

Trees; Michael Drmota


Basic Notions

Generating Functions

Unlabeled Trees

Labeled Trees

Selected Topics on Trees

Planar maps; Gilles Schaeffer

What is a Map?

Counting Tree-Rooted Maps

Counting Planar Maps

Beyond Planar Maps, an Even Shorter Account

Graph Enumeration; Marc Noy


Graph Decompositions

Connected Graphs with Given Excess

Regular Graphs

Monotone and Hereditary Classes

Planar Graphs

Graphs on Surfaces and Graph Minors


Unlabelled Graphs

Unimodality, Log-Concavity, Real–Rootedness and Beyond; Petter Brándén


Probabilistic Consequences of Real–Rootedness

Unimodality and G-Nonnegativity

Log–Concavity and Matroids

Infinite Log-Concavity

The Neggers–Stanley Conjecture

Preserving Real–Rootedness

Common Interleavers

Multivariate Techniques

Historical Notes

Words; Dominique Perrin and Antonio Restivo




Lyndon words

Eulerian Graphs and De Bruijn Cycles

Unavoidable Sets

The Burrows-Wheeler Transform

The Gessel-Reutenauer Bijection

Suffix Arrays

Tilings; James Propp

Introduction and Overview

The Transfer Matrix Method

Other Determinant Methods

Representation-Theoretic Methods

Other Combinatorial Methods

Related Topics, and an Attempt at History

Some Emergent Themes



Lattice Path Enumeration; Christian Krattenthaler


Lattice Paths Without Restrictions

Linear Boundaries of Slope 1

Simple Paths with Linear Boundaries of Rational Slope, I

Simple Paths with Linear Boundaries with Rational Slope, II

Simple Paths with a Piecewise Linear Boundary

Simple Paths with General Boundaries

Elementary Results on Motzkin and Schroder Paths

A continued Fraction for the Weighted Counting of Motzkin Paths

Lattice Paths and Orthogonal Polynomials

Motzkin Paths in a Strip

Further Results for Lattice Paths in the Plane

Non-Intersecting Lattice Paths

Lattice Paths and Their Turns

Multidimensional Lattice Paths

Multidimensional Lattice Paths Bounded by a Hyperplane

Multidimensional Paths With a General Boundary

The Reflection Principle in Full Generality

Q-Counting Of Lattice Paths and Rogers–Ramanujan Identities

Self-Avoiding Walks

Catalan Paths and q; t-enumeration; James Haglund

Introduction to q-Analogues and Catalan Numbers

The q; t-Catalan Numbers

Parking Functions and the Hilbert Series

The q; t-Schrӧder Polynomial

Rational Catalan Combinatorics

Permutation Classes; Vincent Vatter


Growth Rates of Principal Classes

Notions of Structure

The Set of All Growth Rates

Parking Functions; Catherine H. Yan


Parking Functions and Labeled Trees

Many Faces of Parking Functions

Generalized Parking Functions

Parking Functions Associated with Graphs

Final Remarks

Standard Young Tableaux; Ron Adin and Yuval Roichman



Formulas for Thin Shapes

Jeu de taquin and the RS Correspondence

Formulas for Classical Shapes

More Proofs of the Hook Length Formula

Formulas for Skew Strips

Truncated and Other Non-Classical Shapes

Rim Hook and Domino Tableaux


Counting Reduced Words

Appendix 1: Representation Theoretic Aspects

Appendix 2: Asymptotics and Probabilistic Aspects

Computer Algebra; Manuel Kauers


Computer Algebra Essentials

Counting Algorithms

Symbolic Summation

The Guess-and-Prove Paradigm


About the Editor

Miklós Bóna received his Ph.D. in mathematics at Massachusetts Institute of Technology in 1997. Since 1999, he has taught at the University of Florida, where in 2010 he was inducted in the Academy of Distinguished Teaching Scholars. He is the author of four books and more than 65 research articles, mostly focusing on enumerative and analytic combinatorics. His book Combinatorics of Permutations won the Outstanding Title Award from Choice, the journal of the American Library Association. He has mentored numerous graduate and undergraduate students. Miklós Bóna is an editor-in-chief for the Electronic Journal of Combinatorics, and for two book series at CRC Press.

About the Series

Discrete Mathematics and Its Applications

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Algebra / General
MATHEMATICS / Number Theory
MATHEMATICS / Combinatorics