Introduction to Enumerative and Analytic Combinatorics: 2nd Edition (Hardback) book cover

Introduction to Enumerative and Analytic Combinatorics

2nd Edition

By Miklos Bona

Chapman and Hall/CRC

534 pages | 147 B/W Illus.

CHOICE 2018 Outstanding Academic Title Award Winner
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Description

Introduction to Enumerative and Analytic Combinatorics fills the gap between introductory texts in discrete mathematics and advanced graduate texts in enumerative combinatorics. The book first deals with basic counting principles, compositions and partitions, and generating functions. It then focuses on the structure of permutations, graph enumeration, and extremal combinatorics. Lastly, the text discusses supplemental topics, including error-correcting codes, properties of sequences, and magic squares.

Strengthening the analytic flavor of the book, this Second Edition:

  • Features a new chapter on analytic combinatorics and new sections on advanced applications of generating functions
  • Demonstrates powerful techniques that do not require the residue theorem or complex integration
  • Adds new exercises to all chapters, significantly extending coverage of the given topics

Introduction to Enumerative and Analytic Combinatorics, Second Edition makes combinatorics more accessible, increasing interest in this rapidly expanding field.

Outstanding Academic Title of the Year, Choice magazine, American Library Association.

Reviews

Bona's work is a superb text for any reader learning the vast topic of combinatorics. It includes a well-written description of the fundamentals of combinatorics and several chapters of applications. Each chapter concludes with a list of important formulas available for future reference and a lengthy list of exercises. These exercises are quite comprehensive in that they include a wide range of topics, many exploring other interesting topics unexplained in the text. Most of these exercises are accompanied by complete, well-explained solutions to assist struggling readers. One of the best aspects of the book is the conversational tone in which it is written. When reading through the numerous proofs in the text, readers will feel as though they are actually in the classroom with Bona (Univ. of Florida). His explanations are clear and concise, and his dry humor is both entertaining and essential to the text's development. People spreading rumors, wearing colorful hats, and embarking on hazardous vacations are much more enjoyable to count than indistinguishable balls in jars. This work is an excellent addition to the combinatorics library.

--A. Misseldine, Southern Utah University

Table of Contents

METHODS

Basic methods

When we add and when we subtract

When we multiply

When we divide

Applications of basic counting principles

The pigeonhole principle

Notes

Chapter review

Exercises

Solutions to exercises

Supplementary exercises

Applications of basic methods

Multisets and compositions

Set partitions

Partitions of integers

The inclusion-exclusion principle

The twelvefold way

Notes

Chapter review

Exercises

Solutions to exercises

Supplementary exercises

Generating functions

Power series

Warming up: Solving recurrence relations

Products of generating functions

Compositions of generating functions

A different type of generating functions

Notes

Chapter review

Exercises

Solutions to exercises

Supplementary exercises

TOPICS

Counting permutations

Eulerian numbers

The cycle structure of permutations

Cycle structure and exponential generating functions

Inversions

Advanced applications of generating functions to permutation enumeration

Notes

Chapter review

Exercises

Solutions to exercises

Supplementary exercises

Counting graphs

Trees and forests

Graphs and functions

When the vertices are not freely labeled

Graphs on colored vertices

Graphs and generating functions

Notes

Chapter review

Exercises

Solutions to exercises

Supplementary exercises

Extremal combinatorics

Extremal graph theory

Hypergraphs

Something is more than nothing: Existence proofs

Notes

Chapter review

Exercises

Solutions to exercises

Supplementary exercises

AN ADVANCED METHOD

Analytic combinatorics

Exponential growth rates

Polynomial precision

More precise asymptotics

Notes

Chapter review

Exercises

Solutions to exercises

Supplementary exercises

SPECIAL TOPICS

Symmetric structures

Designs

Finite projective planes

Error-correcting codes

Counting symmetric structures

Notes

Chapter review

Exercises

Solutions to exercises

Supplementary exercises

Sequences in combinatorics

Unimodality

Log-concavity

The real zeros property

Notes

Chapter review

Exercises

Solutions to exercises

Supplementary exercises

Counting magic squares and magic cubes

A distribution problem

Magic squares of fixed size

Magic squares of fixed line sum

Why magic cubes are different

Notes

Chapter review

Exercises

Solutions to exercises

Supplementary exercises

Appendix: The method of mathematical induction

Weak induction

Strong induction

About the Author

Miklós Bóna received his Ph.D in mathematics from the Massachusetts Institute of Technology in 1997. Since 1999, he has taught at the University of Florida, where, in 2010, he was inducted into the Academy of Distinguished Teaching Scholars. Professor Bóna has mentored numerous graduate and undergraduate students. 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 a 2006 Outstanding Title Award from Choice, the journal of the American Library Association. He is also 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:
COM046000
COMPUTERS / Operating Systems / General
MAT000000
MATHEMATICS / General
MAT036000
MATHEMATICS / Combinatorics