## A Course in EnumerationCombinatorial 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. |

### What people are saying - Write a review

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

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 |

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 |

557 | |