# A Course in Enumeration

Springer Science & Business Media, Jun 28, 2007 - Mathematics - 565 pages

Combinatorial enumeration is a readily accessible subject full of easily stated, but sometimes tantalizingly difficult problems. This book leads the reader in a leisurely way from the basic notions to a variety of topics, ranging from algebra to statistical physics. Its aim is to introduce the student to a fascinating field, and to be a source of information for the professional mathematician who wants to learn more about the subject. The book is organized in three parts: Basics, Methods, and Topics. There are 666 exercises, and as a special feature every chapter ends with a highlight, discussing a particularly beautiful or famous result.

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An Amazing book. I enjoyed it a lot. I would, however, not recommend it for someone who is going to use it only without any other material because the excercises are a lot tougher. Overall, its a good book for graduate students and post doctoral students who need a serious refresher in combinatorics.

### Contents

 Fundamental Coefficients 5 Exercises 9 12 Subsets and Binomial Coefficients 10 Exercises 18 13 Setpartitions and Stirling Numbers S𝘯𝘬 20 Exercises 23 14 Permutations and Stirling Numbers s𝘯𝘬 24 Exercises 29
 64 Symmetries on N and R 270 Exercises 276 Patterns of Polyominoes 278 Notes and References 285 The Catalan Connection 289 71 Catalan Matrices and Orthogonal Polynomials 290 Exercises 297 72 Catalan Numbers and Lattice Paths 300

 15 NumberPartitions 31 Exercises 35 16 Lattice Paths and Gaussian Coefficients 36 Exercises 42 Aztec Diamonds 44 Notes and References 51 Formal Series and Infinite Matrices 53 Exercises 59 22 Types of Formal Series 60 Exercises 65 23 Infinite Sums and Products 66 Exercises 70 24 Infinite Matrices and Inversion of Sequences 71 Exercises 76 25 Probability Generating Functions 77 Exercises 84 The Point of No Return 85 Notes and References 90 Generating Functions 93 Exercises 102 32 Evaluating Sums 105 Exercises 110 33 The Exponential Formula 112 Exercises 122 34 NumberPartitions and Infinite Products 124 Exercises 132 Ramanujans Most Beautiful Formula 136 Notes and References 141 Hypergeometric Summation 143 Exercises 148 Exercises 155 Exercises 161 44 Hypergeometric Series 162 Exercises 168 New Identities from Old 171 Notes and References 178 Sieve Methods 179 Exercises 189 52 Möbius Inversion 191 Exercises 200 53 The Involution Principle 202 Exercises 215 54 The Lemma of GesselViennot 217 Exercises 229 Tuttes MatrixTree Theorem 231 Notes and References 237 Enumeration of Patterns 239 Exercises 248 62 The Theorem of PólyaRedfield 249 Exercises 260 63 Cycle Index 262 Exercises 269
 Exercises 305 73 Generating Functions and Operator Calculus 306 Exercises 320 74 Combinatorial Interpretation of Catalan Numbers 323 Exercises 333 Chord Diagrams 337 Notes and References 344 Symmetric Functions 345 Exercises 349 82 Homogeneous Symmetric Functions 350 Exercises 355 83 Schur Functions 356 Exercises 366 84 The RSK Algorithm 367 Exercises 378 85 Standard Tableaux 380 Exercises 383 HookLength Formulas 385 Notes and References 391 Counting Polynomials 393 Exercises 405 92 Eulerian Cycles and the Interlace Polynomial 407 Exercises 419 93 Plane Graphs and Transition Polynomials 420 Exercises 432 94 Knot Polynomials 434 Exercises 443 The BEST Theorem 445 Notes and References 449 Models from Statistical Physics 451 Exercises 465 102 The Ising Problem and Eulerian Subgraphs 467 Exercises 480 103 Hard Models 481 Exercises 489 104 Square Ice 490 Exercises 504 The RogersRamanujan Identities 506 Notes and References 517 Solutions to Selected Exercises 519 Chapter 2 521 Chapter 3 524 Chapter 4 528 Chapter 5 529 Chapter 6 533 Chapter 7 536 Chapter 8 540 Chapter 9 544 Chapter 10 547 Notation 552 Index 557 Copyright