A Course in Enumeration

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

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 M÷bius 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 PˇlyaRedfield
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
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