Introduction to Enumerative Combinatorics (Walter Rudin Student Series in Advanced Mathematics)

544 Pages·2007·4.71 MB·English

by  Miklos Bona

Introduction to
Enumerative
Combinatorics
Miklos Bona
Mc
Graw Higher Education
Hill
Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis
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(cid:9)
The McGraw Hill Companies
Mc
Graw Higher Education
Hill
INTRODUCTION TO ENUMERATIVE COMBINATORICS
Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the
Americas, New York, NY 10020. Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights
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1 2 3 4 5 6 7 8 9 0 DOC/DOC 0 9 8 7 6 5
ISBN-13 978-0-07-312561-9
ISBN-10 0-07-312561–X
Publisher: Elizabeth J. Haefele
Senior Sponsoring Editor: Elizabeth Covello
Developmental Editor: Dan Seibert
Senior Marketing Manager: Nancy Anselment Bradshaw
Project Manager: April R. Southwood
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Cover Illustration: Rokusek Design
Compositor: Lachina Publishing Services
Typeface: 11/13 NewTimes Roman
Printer: R. R. Donnelley Crawfordsville, IN
Library of Congress Cataloging-in-Publication Data
B6na, MiklOs.
Introduction to enumerative combinatorics / Bona, Miklos. — 1st ed.
p.(cid:9) cm.
Includes bibliographical references and index.
ISBN 978-0-07-312561-9 — ISBN 0-07-312561–X (acid-free paper)
1. Combinatorial analysis—Textbooks. 2. Combinatorial enumeration problems—Textbooks. I. Title.
QA164.8.B66 2007
(cid:9)
511'.6—dc22 2005050498
CIP
www.mhhe.com
To Linda
To Mikike, Benny, and Vinnie
Titles in the Walter Rudin Student Series in Advanced Mathematics
Bona, Miklos
Introduction to Enumerative Combinatorics
Chartrand, Gary and Ping Zhang
Introduction to Graph Theory
Davis, Sheldon
Topology
Dumas, Bob and John E. McCarthy
Transition to Higher Mathematics: Structure and Proof
(publishing in Spring 2006)
Rudin, Walter
Functional Analysis, 2'd Edition
Rudin, Walter
Principles of Mathematical Analysis, 3rd Edition
Rudin, Walter
Real and Complex Analysis, 3rd Edition
Simmons, George F. and Steven G. Krantz
Differential Equations: Theory, Technique, and Practice
(publishing in Spring 2006)
Walter Rudin Student Series in Advanced Mathematics - Editorial Board
Editor-in-Chief: Steven G. Krantz, Washington University in St. Louis
David Barrett Jean Pierre Rosay
-
University of Michigan University of Wisconsin
Steven Bell Jonathan Wahl
Purdue University University of North Carolina
John P. D'Angelo Lawrence Washington
University of Illinois at Urbana-Champaign University of Maryland
Robert Fefferman C. Eugene Wayne
University of Chicago Boston University
William McCallum Michael Wolf
University of Arizona Rice University
Bruce Palka HungHsi Wu
-
University of Texas at Austin University of California, Berkeley
Harold R. Parks
Oregon State University
Contents
Foreword(cid:9) xi
Preface(cid:9) xiii
Acknowledgments(cid:9) xv
I How: Methods(cid:9) 1
1 Basic Methods(cid:9) 3
1.1 When We Add and When We Subtract (cid:9) 3
1.1.1 When We Add (cid:9) 3
1.1.2 When We Subtract (cid:9) 4
1.2 When We Multiply (cid:9) 6
1.2.1 The Product Principle (cid:9) 6
1.2.2 Using Several Counting Principles (cid:9) 9
1.2.3 When Repetitions Are Not Allowed(cid:9) 10
1.3 When We Divide (cid:9) 14
1.3.1 The Division Principle (cid:9) 14
1.3.2 Subsets (cid:9) 17
1.4 Applications of Basic Counting Principles (cid:9) 20
1.4.1 Bijective Proofs (cid:9) 20
1.4.2 Properties of Binomial Coefficients (cid:9) 27
1.4.3 Permutations With Repetition (cid:9) 31
1.5 The Pigeonhole Principle (cid:9) 35
1.6 Notes (cid:9) 39
1.7 Chapter Review (cid:9) 40
1.8 Exercises (cid:9) 41
1.9 Solutions to Exercises (cid:9) 46
1.10 Supplementary Exercises (cid:9) 54
vi Contents
2 Direct Applications of Basic Methods(cid:9) 59
2.1 Multisets and Compositions (cid:9) 59
2.1.1(cid:9) Weak Compositions 59
2.1.2(cid:9) Compositions (cid:9) 62
2.2 Set Partitions (cid:9) 63
2.2.1(cid:9) Stirling Numbers of the Second Kind (cid:9) 63
2.2.2(cid:9) Recurrence Relations for Stirling Numbers of the
Second Kind (cid:9) 65
2.2.3(cid:9) When the Number of Blocks Is Not Fixed (cid:9) 69
2.3 Partitions of Integers (cid:9) 70
2.3.1(cid:9) Nonincreasing Finite Sequences of Integers 70
2.3.2(cid:9) Ferrers Shapes and Their Applications (cid:9) 72
2.3.3(cid:9) Excursion: Euler's Pentagonal Number Theorem 75
2.4 The Inclusion-Exclusion Principle (cid:9) 83
2.4.1(cid:9) Two Intersecting Sets (cid:9) 83
2.4.2(cid:9) Three Intersecting Sets (cid:9) 86
2.4.3(cid:9) Any Number of Intersecting Sets (cid:9) 90
2.5 The Twelvefold Way (cid:9) 99
2.6 Notes (cid:9) 102
2.7 Chapter Review (cid:9) 103
2.8 Exercises (cid:9) 104
2.9 Solutions to Exercises (cid:9) 108
2.10 Supplementary Exercises (cid:9) 120
3 Generating Functions 125
3.1 Power Series (cid:9) 125
3.1.1(cid:9) Generalized Binomial Coefficients (cid:9) 125
3.1.2(cid:9) Formal Power Series (cid:9) 127
3.2 Warming Up: Solving Recursions (cid:9) 130
3.2.1(cid:9) Ordinary Generating Functions (cid:9) 130
3.2.2(cid:9) Exponential Generating Functions (cid:9) 138
3.3 Products of Generating Functions (cid:9) 141
3.3.1(cid:9) Ordinary Generating Functions (cid:9) 142
3.3.2(cid:9) Exponential Generating Functions (cid:9) 154
3.4 Excursion: Composition of Two Generating Functions 160
3.4.1(cid:9) Ordinary Generating Functions (cid:9) 160
3.4.2(cid:9) Exponential Generating Functions (cid:9) 165
3.5 Excursion: A Different Type of Generating Function 173
3.6 Notes (cid:9) 174
3.7 Chapter Review (cid:9) 175
Contents vii
3.8(cid:9) Exercises (cid:9) 176
3.9(cid:9) Solutions to Exercises (cid:9) 179
3.10 Supplementary Exercises (cid:9) 190
II(cid:9) What: Topics 193
4 Counting Permutations 195
4.1 Eulerian Numbers (cid:9) 195
4.2 The Cycle Structure of Permutations (cid:9) 204
4.2.1(cid:9) Stirling Numbers of the First Kind (cid:9) 204
4.2.2(cid:9) Permutations of a Given Type (cid:9) 212
4.3 Cycle Structure and Exponential Generating Functions . 217
4.4 Inversions (cid:9) 222
4.4.1(cid:9) Counting Permutations with Respect to Inversions 227
4.5 Notes (cid:9) 232
4.6 Chapter Review (cid:9) 233
4.7 Exercises (cid:9) 234
4.8 Solutions to Exercises (cid:9) 239
4.9 Supplementary Exercises (cid:9) 251
5 Counting Graphs 255
5.1 Counting Trees and Forests (cid:9) 258
5.1.1(cid:9) Counting Trees (cid:9) 258
5.2 The Notion of Graph Isomorphisms (cid:9) 260
5.3 Counting Trees on Labeled Vertices (cid:9) 265
5.3.1(cid:9) Counting Forests (cid:9) 274
5.4 Graphs and Functions (cid:9) 278
5.4.1(cid:9) Acyclic Functions (cid:9) 278
5.4.2(cid:9) Parking Functions (cid:9) 279
5.5 When the Vertices Are Not Freely Labeled (cid:9) 283
5.5.1(cid:9) Rooted Plane Trees (cid:9) 283
5.5.2(cid:9) Binary Plane Trees (cid:9) 288
5.6 Excursion: Graphs on Colored Vertices (cid:9) 292
5.6.1(cid:9) Chromatic Polynomials (cid:9) 294
5.6.2(cid:9) Counting k-colored Graphs (cid:9) 301
5.7 Graphs and Generating Functions (cid:9) 305
5.7.1(cid:9) Generating Functions of Trees (cid:9) 305
5.7.2(cid:9) Counting Connected Graphs 306
5.7.3(cid:9) Counting Eulerian Graphs (cid:9) 307
viii Contents
5.8(cid:9) Notes (cid:9) 311
5.9(cid:9) Chapter Review (cid:9) 313
5.10 Exercises (cid:9) 314
5.11 Solutions to Exercises (cid:9) 319
5.12 Supplementary Exercises (cid:9) 330
6 Extremal Combinatorics 335
6.1 Extremal Graph Theory (cid:9) 335
6.1.1(cid:9) Bipartite Graphs (cid:9) 335
6.1.2(cid:9) Turan's Theorem (cid:9) 340
6.1.3(cid:9) Graphs Excluding Cycles (cid:9) 344
6.1.4(cid:9) Graphs Excluding Complete Bipartite Graphs . . 354
6.2 Hypergraphs (cid:9) 356
6.2.1(cid:9) Hypergraphs with Pairwise Intersecting Edges 357
6.2.2(cid:9) Hypergraphs with Pairwise Incomparable Edges 364
6.3 Something Is More Than Nothing: Existence Proofs 366
6.3.1(cid:9) Property B (cid:9) 367
6.3.2(cid:9) Excluding Monochromatic Arithmetic Progressions 368
6.3.3(cid:9) Codes Over Finite Alphabets (cid:9) 369
6.4 Notes (cid:9) 373
6.5 Chapter Review (cid:9) 374
6.6 Exercises (cid:9) 375
6.7 Solutions to Exercises (cid:9) 381
6.8 Supplementary Exercises (cid:9) 393
III What Else: Special Topics 397
7 Symmetric Structures 399
7.1 Hypergraphs with Symmetries (cid:9) 399
7.2 Finite Projective Planes (cid:9) 406
7.2.1(cid:9) Excursion: Finite Projective Planes of Prime Power
Order (cid:9) 409
7.3 Error-Correcting Codes 411
7.3.1(cid:9) Words Far Apart (cid:9) 411
7.3.2(cid:9) Codes from Hypergraphs 414
7.3.3(cid:9) Perfect Codes (cid:9) 415
7.4 Counting Symmetric Structures (cid:9) 418
7.5 Notes (cid:9) 427
7.6 Chapter Review (cid:9) 428
Contents(cid:9) ix
7.7 Exercises (cid:9) 429
7.8 Solutions to Exercises (cid:9) 430
7.9 Supplementary Exercises (cid:9) 435
8 Sequences in Combinatorics(cid:9) 439
8.1 Unimodality (cid:9) 439
8.2 Log-Concavity (cid:9) 442
8.2.1 Log-Concavity Implies Unimodality(cid:9) 442
8.2.2 The Product Property (cid:9) 445
8.2.3 Injective Proofs (cid:9) 447
8.3 The Real Zeros Property (cid:9) 453
8.4 Notes (cid:9) 457
8.5 Chapter Review (cid:9) 458
8.6 Exercises (cid:9) 458
8.7 Solutions to Exercises (cid:9) 460
8.8 Supplementary Exercises (cid:9) 466
9 Counting Magic Squares and Magic Cubes(cid:9) 469
9.1 An Interesting Distribution Problem (cid:9) 469
9.2 Magic Squares of Fixed Size (cid:9) 470
9.2.1 The Case of n = 3 (cid:9) 471
9.2.2 The Function Hn ( r ) for Fixed n(cid:9) 474
9.3 Magic Squares of Fixed Line Sum (cid:9) 485
9.4 Why Magic Cubes Are Different(cid:9) 490
9.5 Notes (cid:9) 493
9.6 Chapter Review (cid:9) 495
9.7 Exercises (cid:9) 496
9.8 Solutions to Exercises (cid:9) 499
9.9 Supplementary Exercises (cid:9) 509
A The Method of Mathematical Induction(cid:9) 511
A.1 Weak Induction (cid:9) 511
A.2 Strong Induction (cid:9) 513
Bibliography(cid:9) 515
Index(cid:9) 521
Frequently Used Notation(cid:9) 525