!!! Aplied Combinatorial Mathematics

8/16/2019 !!! Aplied Combinatorial Mathematics

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APPLIED COMBINATORIAL

MATHEMATICS

The Authors

GEORGE POLYA

DERRICK H. L E H ME R

MONTGOMERY PHISTER, Jr.

JOHN RIORDAN

ELLIOTT W. MONTROLL

N. G. D E BRUIJN

FRANK HARARY

RICHARD BELLMAN

ROBERT KALABA

EDWIN L. PETERSON

LEO BREIMAN

ALBERTW.

TUCKER

EDWIN F. BECKENBACH

MARSHALL HALL, Jr.

JACOB WOLFOWITZ

CHARLES B. TOMPKINS

KENNETH N. TRUEBLOOD

GEORGE GAMOW

HERMANN WE Y L

Editor

EDWIN F. BECKENBACHProfessor of MathematicsUniversity of CaliforniaLos Angeles

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CONTENTS

INTRODUCTION 1

PART 1A COMPUTATION AND EVALUATION

chapter 1 THE MACHINE TOOLS OF COMBINATORICS 5DERRICK H. LEHMER

1.1 Introduction 5

1.2 Representation and Processing of Digital Information 6

1.3 Application of Representations 11

1.4 Signatures and Their Orderly Generation 17

1.5 Orderly Listing of Permutations 19

1.6 Orderly Listing of Combinations 24

1.7 Orderly Listing of Compositions 24

1.8 Orderly Listing of Partitions 25

1.9 The Back-Track Procedure 26

1.10 Ranking of Combinations 27

chapter 2 TECHNIQUES FOR SIMPLIFYING LOGICAL

NETWORKS 32MONTGOMERY PHISTER, JR.

2.1 Introduction 32

Simplifying a Given Sequential Circuit 37

2.2 Simplest AND-OR Gates 37

2.3 Multiple-Output Function Simplifications 40

2.4 Elimination of Memory Elements 42

2.5 Alternative Methods of Assigning Flip-Flop States 49

C/ioosing Decision and Memory Elements 53

2.6 Memory Elements Other Than Delay Flip-Flops 53

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xvi Contents

2.7 Decision Elements Other Than AND and OR Gates

2.8 Conclusion

57

61

PART IB COUNTING AND ENUMERATION

chapter 3 GENERATING FUNCTIONS 67JOHN RIORDAN

3.1 Introduction 67

3.2 Generating Functions for Combinations 69

3.3 Generating Functions for Permutations 71

3.4 Elementary Relations: Ordinary Generating Function 72

3.5 Elementary Relations: Exponential Generating Function 79

3.6 Generating Functions in Probability and Statistics 84

3.7 Polya s Enumeration Theorem 90

chapter 4 LATTICE STATISTICS 96ELLIOTT W. MONTROLL

4.1 Introduction 96

4.2 Random Walks on Lattices and the Polya Problem 97

4.3 More General Random Walks on Lattices 103

4.4 The PfafRan and the Dimer Problem 105

4.5 Cyclic Matrices 113

4.6 Evaluation of Dimer Pfaffian 116

4.7 The Ising Problem 121

4.8 Some Remarks on Periodic Boundary Conditions 132

4.9 Lattice Statistics of Slightly Defective Lattices 134

chapter 5 POLYA S THEORY OF COUNTING 144

N. G. DE BRUIJN

5.1 Introduction 144

5.2 Cycle Index of a Permutation Group 145

5.3 The Main Lemma 150

5.4 Functions and Patterns 151

5.5 Weight of a Function; Weight of a Pattern 153

5.6 Store and Inventory 154

5.7 Inventory of a Function 155

5.8 The Pattern Inventory; Polya s Theorem 157

5.9 Generalization of Polya s Theorem 161

5.10 Patterns of One-to-One Mappings 164

5.11 Labeling and Delabeling 168


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Contents xvii

5.12 The Total Number of Patterns 171

5.13 The Kranz Group 176

5.14 Epilogue 180

chapter 6 COMBINATORIAL PROBLEMS IN GRAPHICALENUMERATION 185

FRANK HARARY


6.2 Graphical Preliminaries 185

6.3 Some Unsolved Problems 189

6.4 Problems Involving Directed Graphs 189

6.5 Problems Involving Partitions 190

6.6 Topological Problems 191

6.7 Problems Involving Connectivity 193

6.8 Problems Involving Groups 195

6.9 Electrical Problems 197

6.10 Physical Problems 199

6.11 Graph-Counting Methods 202

6.12 Tree-Counting Methods 205

6.13 Comparison of Solved and Unsolved Problems 208

6.14 Some Applications of Enumeration Problems to Other Fields 212

PART 2 CONTROL AND EXTREMIZATION

chapter 7 DYNAMIC PROGRAMMING AND MARKOVIANDECISION PROCESSES, WITH PARTICULAR

APPLICATION TO BASEBALL AND CHESS 221RICHARD BELLMAN


7.2 Baseball as a Multistage Decision Process; State Variables 221

7.3 Decisions 222

7.4 Criterion Function 223

7.5 Policies, Optimum Policies/ and Combinatorics 224

7.6 Principle of Optimality 226

7.7 Functional Equations 226

7.8 Existence and Uniqueness 227

7.9 Analytic Aspects 227

7.10 Direct Computational Approach 229

7.11 Stratification 229

7.12 Approximation in Policy Space 230


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xviii Contents

7.13 Utilization of Experience an d In tuition 231

7 14 Two-Team Version of Baseball 232

7.15 Probability of Tying or Winning 232

7.16 Inventory and Replacement Processes and Asymptotic Behavior 2327.17 Chess as a Multistage Decision Process 233

7.18 Computational Aspects 234

chapter 8 GRAPH THEORY AND AUTOMATIC CONTROL 237ROBERT KALABA


8.2 Time-Optimal Control and the Nature of Feedback 238

8.3 Formulation 239

8.4 Uniqueness 240

8.5 Successive Approximations 241

8.6 Observations on the Approximation Scheme 242

8.7 Other Approaches 242

8.8 Arbitrary Terminal States 243

8.9 Preferred Suboptima l Trajectories 243

8.10 A Stochastic Time-Optimal Control Process 244

8.11 Minimax Control Processes 246

8.12 Use of Functional Equations 246

8.13 A Special Case 246

8.14 Minimal Spanning Trees 247

8.15 Comments and Interconnections 248

8.16 Multiple Stresses 249

8.17 Discussion 250

chapter 9 OPTIMUM MULT1VARIABLE CONTROL 253EDWIN L. PETERSON

9.1 Problem Structure 253

9.2 Alternative Approaches 257

9.3 Polynomial Approximation 264

9.4 Linearization and Successive Approximation 272

9.5 Physical Implementation 279

chapter 10 STOPPING-RULE PR O B LEMS 284LEO BREIMAN

10.1 How to Recognize a Stopping-Rule Problem 284

10.2 Examples 285

10.3 Formulation 287

10.4 What Is a Stopping Rule? 290


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Contents xix

10.5 What Is a Solution? 291

10.6 Stop When You Are Ahead; the Stability Problem 292

10.7 The Functional Equation 294

10.8 Elimination of Forced Continuation 296

10.9 Entrance-Fee Problems and Reduction to Them 298

10.10 The Linear-Programming Solution 302

Binary Decision Renewal Problems 305

10.11 Introduction and Examples 305

10.12 Formulation 306

10.13 What Is a Solution? 307

10.14 Reduction to a Stopping-Rule Problem by Cycle Analysis 308

10.15 A Direct Approach 312

10.16 The Linear-Programming Solution 314

10.17 A Duality Relation 317

chapter COMBINATORIAL A L G E B R A OF MATRIX GAMESAND LINEAR PROGRAMS 320ALBERT W. TUCKER


11.2 Game Example 320

11.3 Key Equation and Basic Solutions 322

11.4 Schematic Representation 324

11.5 Equivalence by Pivot Steps; Simplex Method 327

11.6 Linear Programming Examples 331

11.7 Key Equation and Basic Solutions for Linear Programs 333

11.8 Canonical Forms 336

11.9 Inverse Basis Procedure 340

11.10 General Formulations 343

chapter 12 NETWORK FLOW PROBLEMS 348

EDWIN F. BECKENBACH


12.2 Networks 348

12.3 Cuts 350

12.4 Flows 351

12.5 Flows and Cuts 352

12.6 Bounds on Capacities of Cuts and on Values of Flows 355

12.7 The Max-Flow—Min-Cut Theorem 357

12.8 The Ford-Fulkerson Algorithm 358

12.9 Example 359

12.10 Extensions and Applications 361

12.11 Graph Theory and Combinatorial Problems 362


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Contents

PART 3 CONSTRUCTION AND EXISTENCE

chapter 13 BLOCK DESIGNS 369

MARSHALL HALL, JR.13.1 Introduction 369

13.2 Block Designs and Latin Squares; Examples and Definitions 370

13.3 Applications of Block Designs 373

13.4 General Theory of Block Designs 377

13.5 Construction of Block Designs and Orthogonal Latin Squares 387

chapter 14 INTRODUCTION TO INFORMATION THEORY 406

JACOB WOLFOWITZ


14.2 Channels 406

14.3 Codes 407

14.4 Entropy; Generated Sequences 408

14.5 The Discrete Memoryless Channel 410

14.6 Compound Channels 412

14.7 Other Channels 414

chapter 15 SPERNER S LEMMA AND SOME EXTENSIONS 416CHARLES B. TOMPKINS


15.2 Sperner s Lemma 418

15.3 Brouwer s Fixed-Point Theorem 424

15.4 Subdivisions of a Simplex 427

15.5 Review of Position—Second Introduction 430

15.6 Oriented Simplexes, Their Boundaries/ and Oriented Simplicial

Mappings 434

15.7 Homology—fc-Chains, Cycles/ Bounding Cycles 437

15.8 The Fundamental Index Theorem 441

15.9 Some Applications 445

15.10 Toward Greater Rigor 453

chapter 16 CRYSTALLOGRAPHY 456KENNETH N. TRUEBLOOD


Nature of the Crystalline Sfate 457

16.2 Lattices 457


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Contents xxi

16.3 Point Symmetry 461

16.4 Space Symmetry 465

16.5 Antisymmetry 467

Crystal Structure Analysis by X Ray Diffraction 468

16.6 Diffraction468

16.7 Fourier Transforms of Atoms and Groups of Atoms 474

16.8 The Phase Problem 478

Approaches to Solving the Phase Problem 482

16.9 The Patterson Function 482

16.10 Heavy-Atom Methods 485

16.11 Isomorphous Replacement 488

16.12 Direct Methods 491

16.13 Refinement 495

Some Aspects of Atomic Arrangements 500

16.14 Topology and Shapes of Molecules and Assemblies of Molecules 500

16.15 Packing of Spheres, Molecules, and Polyhedra 504

16.16 Disordered Structures and Order-Disorder Phenomena 507

chapter 17 COMBINATORIAL PRINCIPLES IN GENETICS 515

GEORGE GAMOW


17.2 Amino-Acid Sequences in Proteins 515

17.3 Double-Stranded Sequences in DNA 519

17.4 Combinatorial Principles 522

17.5 The Overlapping-Code Hypothesis 522

17.6 Statistical Investigations 523

17.7 Geometric Implications 525

17.8 Statistical Assignments 526

17.9 Monte Carlo Methods 528

17.10 Experimental Results 532

chapter 18 APPENDICES 536HERMANN W E Y L

18.1 Ars Combinatoria 536

18.2 Quantum Physics and Causality 551

18.3 Chemical Valence and the Hierarchy of Structures 563

18.4 Physics and Biology 572

ANSWERS TO MULTIPLE-CHOICE REVIEW P ROBLEMS 583

AUTHOR INDEX 585

SUBJECT INDEX 591

!!! Aplied Combinatorial Mathematics

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