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Introduction to Enumerative Combinatorics [Kõva köide]

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  • Formaat: Hardback, 544 pages, kõrgus x laius x paksus: 244x165x30 mm, kaal: 853 g, Illustrations
  • Ilmumisaeg: 16-Oct-2005
  • Kirjastus: McGraw-Hill Professional
  • ISBN-10: 007312561X
  • ISBN-13: 9780073125619
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Introduction to Enumerative CombinatoricsSuurem pilt
  • Formaat: Hardback, 544 pages, kõrgus x laius x paksus: 244x165x30 mm, kaal: 853 g, Illustrations
  • Ilmumisaeg: 16-Oct-2005
  • Kirjastus: McGraw-Hill Professional
  • ISBN-10: 007312561X
  • ISBN-13: 9780073125619
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780073125619.html
Märksõnad:
Written by one of the leading authors and researchers in the field, this comprehensive modern text offers a strong focus on enumeration, a vitally important area in introductory combinatorics crucial for further study in the field. Miklós Bóna's text fills the gap between introductory textbooks in discrete mathematics and advanced graduate textbooks in enumerative combinatorics, and is one of the very first intermediate-level books to focus on enumerative combinatorics. The text can be used for an advanced undergraduate course by thoroughly covering the chapters in Part I on basic enumeration and by selecting a few special topics, or for an introductory graduate course by concentrating on the main areas of enumeration discussed in Part II. The special topics of Part III make the book suitable for a reading course.

This text is part of the Walter Rudin Student Series in Advanced Mathematics.

Foreword xi
Preface xiii
Acknowledgments xv
I How: Methods 1(192)
1 Basic Methods
3(56)
1.1 When We Add and When We Subtract
3(3)
1.1.1 When We Add
3(1)
1.1.2 When We Subtract
4(2)
1.2 When We Multiply
6(8)
1.2.1 The Product Principle
6(3)
1.2.2 Using Several Counting Principles
9(1)
1.2.3 When Repetitions Are Not Allowed
10(4)
1.3 When We Divide
14(6)
1.3.1 The Division Principle
14(3)
1.3.2 Subsets
17(3)
1.4 Applications of Basic Counting Principles
20(15)
1.4.1 Bijective Proofs
20(7)
1.4.2 Properties of Binomial Coefficients
27(4)
1.4.3 Permutations With Repetition
31(4)
1.5 The Pigeonhole Principle
35(4)
1.6 Notes
39(1)
1.7
Chapter Review
40(1)
1.8 Exercises
41(5)
1.9 Solutions to Exercises
46(8)
1.10 Supplementary Exercises
54(5)
2 Direct Applications of Basic Methods
59(66)
2.1 Multisets and Compositions
59(4)
2.1.1 Weak Compositions
59(3)
2.1.2 Compositions
62(1)
2.2 Set Partitions
63(7)
2.2.1 Stirling Numbers of the Second Kind
63(2)
2.2.2 Recurrence Relations for Stirling Numbers of the Second Kind
65(4)
2.2.3 When the Number of Blocks Is Not Fixed
69(1)
2.3 Partitions of Integers
70(13)
2.3.1 Nonincreasing Finite Sequences of Integers
70(2)
2.3.2 Ferrers Shapes and Their Applications
72(3)
2.3.3 Excursion: Euler's Pentagonal Number Theorem
75(8)
2.4 The Inclusion-Exclusion Principle
83(16)
2.4.1 Two Intersecting Sets
83(3)
2.4.2 Three Intersecting Sets
86(4)
2.4.3 Any Number of Intersecting Sets
90(9)
2.5 The Twelvefold Way
99(3)
2.6 Notes
102(1)
2.7
Chapter Review
103(1)
2.8 Exercises
104(4)
2.9 Solutions to Exercises
108(12)
2.10 Supplementary Exercises
120(5)
3 Generating Functions
125(68)
3.1 Power Series
125(5)
3.1.1 Generalized Binomial Coefficients
125(2)
3.1.2 Formal Power Series
127(3)
3.2 Warming Up: Solving Recursions
130(11)
3.2.1 Ordinary Generating Functions
130(8)
3.2.2 Exponential Generating Functions
138(3)
3.3 Products of Generating Functions
141(19)
3.3.1 Ordinary Generating Functions
142(12)
3.3.2 Exponential Generating Functions
154(6)
3.4 Excursion: Composition of Two Generating Functions
160(13)
3.1.1 Ordinary Generating Functions
160(5)
3.4.2 Exponential Generating Functions
165(8)
3.5 Excursion: A Different Type of Generating Function
173(1)
3.6 Notes
174(1)
3.7
Chapter Review
175(1)
3.8 Exercises
176(3)
3.9 Solutions to Exercises
179(11)
3.10 Supplementary Exercises
190(3)
II What: Topics 193(204)
4 Counting Permutations
195(60)
4.1 Eulerian Numbers
195(9)
4.2 The Cycle Structure of Permutations
204(13)
4.2.1 Stirling Numbers of the First Kind
204(8)
4.2.2 Permutations of a Given Type
212(5)
4.3 Cycle Structure and Exponential Generating Functions
217(5)
4.4 Inversions
222(10)
4.4.1 Counting Permutations with Respect to Inversions
227(5)
4.5 Notes
232(1)
4.6
Chapter Review
233(1)
4.7 Exercises
234(5)
4.8 Solutions to Exercises
239(12)
4.9 Supplementary Exercises
251(4)
5 Counting Graphs
255(80)
5.1 Counting Trees and Forests
258(2)
5.1.1 Counting Trees
258(2)
5.2 The Notion of Graph Isomorphisms
260(5)
5.3 Counting Trees on Labeled Vertices
265(13)
5.3.1 Counting Forests
274(4)
5.4 Graphs and Functions
278(5)
5.4.1 Acyclic Functions
278(1)
5.4.2 Parking Functions
279(4)
5.5 When the Vertices Are Not Freely Labeled
283(9)
5.5.1 Rooted Plane Trees
283(5)
5.5.2 Binary Plane Trees
288(4)
5.6 Excursion: Graphs on Colored Vertices
292(13)
5.6.1 Chromatic Polynomials
294(7)
5.6.2 Counting k-colored Graphs
301(4)
5.7 Graphs and Generating Functions
305(6)
5.7.1 Generating Functions of Trees
305(1)
5.7.2 Counting Connected Graphs
306(1)
5.7.3 Counting Eulerian Graphs
307(4)
5.8 Notes
311(2)
5.9
Chapter Review
313(1)
5.10 Exercises
314(5)
5.11 Solutions to Exercises
319(11)
5.12 Supplementary Exercises
330(5)
6 Extremal Combinatorics
335(62)
6.1 Extremal Graph Theory
335(21)
6.1.1 Bipartite Graphs
335(5)
6.1.2 Turan's Theorem
340(4)
6.1.3 Graphs Excluding Cycles
344(10)
6.1.4 Graphs Excluding Complete Bipartite Graphs
354(2)
6.2 Hypergraphs
356(10)
6.2.1 Hypergraphs with Pairwise Intersecting Edges
357(7)
6.2.2 Hypergraphs with Pairwise Incomparable Edges
364(2)
6.3 Something Is More Than Nothing: Existence Proofs
366(7)
6.3.1 Property B
367(1)
6.3.2 Excluding Monochromatic Arithmetic Progressions
368(1)
6.3.3 Codes Over Finite Alphabets
369(4)
6.4 Notes
373(1)
6.5
Chapter Review
374(1)
6.6 Exercises
375(6)
6.7 Solutions to Exercises
381(12)
6.8 Supplementary Exercises
393(4)
III What Else: Special Topics 397(114)
7 Symmetric Structures
399(40)
7.1 Hypergraphs with Symmetries
399(7)
7.2 Finite Projective Planes
406(5)
7.2.1 Excursion: Finite Projective Planes of Prime Power Order
409(2)
7.3 Error-Correcting Codes
411(7)
7.3.1 Words Far Apart
411(3)
7.3.2 Codes from Hypergraphs
414(1)
7.3.3 Perfect Codes
415(3)
7.4 Counting Symmetric Structures
418(9)
7.5 Notes
427(1)
7.6
Chapter Review
428(1)
7.7 Exercises
429(1)
7.8 Solutions to Exercises
430(5)
7.9 Supplementary Exercises
435(4)
8 Sequences in Combinatorics
439(30)
8.1 Unimodality
439(3)
8.2 Log-Concavity
442(11)
8.2.1 Log-Concavity Implies Unimodality
442(3)
8.2.2 The Product Property
445(2)
8.2.3 Injective Proofs
447(6)
8.3 The Real Zeros Property
453(4)
8.4 Notes
457(1)
8.5
Chapter Review
458(1)
8.6 Exercises
458(2)
8.7 Solutions to Exercises
460(6)
8.8 Supplementary Exercises
466(3)
9 Counting Magic Squares and Magic Cubes
469(42)
9.1 An Interesting Distribution Problem
469(1)
9.2 Magic Squares of Fixed Size
470(15)
9.2.1 The Case of n = 3
471(3)
9.2.2 The Function Hn(r) for Fixed n
474(11)
9.3 Magic Squares of Fixed Line Sum
485(5)
9.4 Why Magic Cubes Are Different
490(3)
9.5 Notes
493(2)
9.6
Chapter Review
495(1)
9.7 Exercises
496(3)
9.8 Solutions to Exercises
499(10)
9.9 Supplementary Exercises
509(2)
A The Method of Mathematical Induction 511(4)
A.1 Weak Induction
511(2)
A.2 Strong Induction
513(2)
Bibliography 515(6)
Index 521(4)
Frequently Used Notation 525