2nd Edition

Introduction to Enumerative and Analytic Combinatorics



ISBN 9781482249095
Published October 2, 2015 by Chapman and Hall/CRC
534 Pages 147 B/W Illustrations

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Book 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.

 

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

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Author(s)

Biography

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.

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