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E-raamat: Introduction to Combinatorics

(Southern Illinois University, Carbondale, USA), (Gordon College, Barnesville, Georgia, USA)
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What Is Combinatorics Anyway?

Broadly speaking, combinatorics is the branch of mathematics dealing

with different ways of selecting objects from a set or arranging objects. It

tries to answer two major kinds of questions, namely, counting questions: how many ways can a selection or arrangement be chosen with a particular set of properties; and structural

questions: does there exist a selection or arrangement of objects with a

particular set of properties?

The authors have presented a text for students at all levels of preparation.

For some, this will be the first course where the students see several real proofs.

Others will have a good background in linear algebra, will have completed the calculus

stream, and will have started abstract algebra.

The text starts by briefly discussing several examples of typical combinatorial problems

to give the reader a better idea of what the subject covers. The next

chapters explore enumerative ideas and also probability. It then moves on to

enumerative functions and the relations between them, and generating functions and recurrences.,

Important families of functions, or numbers and then theorems are presented.



Brief introductions to computer algebra and group theory come next. Structures of particular

interest in combinatorics: posets, graphs, codes, Latin squares, and experimental designs follow. The

authors conclude with further discussion of the interaction between linear algebra

and combinatorics.

Features











Two new chapters on probability and posets.





Numerous new illustrations, exercises, and problems.





More examples on current technology use





A thorough focus on accuracy





Three appendices: sets, induction and proof techniques, vectors and matrices, and biographies with historical notes,





Flexible use of MapleTM and MathematicaTM

Arvustused

In Introduction to Combinatorics, Wallis (emer., Southern Illinois Univ.) and George (Gordon State College) present a well-thought-out compilation of topics covering elementary combinatorics. At the beginning, the authors present a thorough background on the fundamentals of combinatorics with topics such as permutations and combinations, the pigeonhole principle, and the principle of inclusion and exclusion. Later chapters are independent of one another and can be selected based on student and instructor interests. These topics include graph theory, coding theory, and balanced incomplete block designs. At the end of each chapter, there are exercises and problems. These vary in difficulty from straightforward practice to more involved proof problems. Solutions and/or hints are provided in the back of the book. In addition, three appendixes discuss proof techniques, matrices and vectors, and historical figures; these allow flexibility in covering the material in various ways that can be based on students backgrounds. Overall, this textbook is a highly readable work that will benefit and enlighten all those interested in learning about combinatorics. It will work in a traditional classroom setting and for independent study. Given the level of material, it is geared toward junior or senior level undergraduate students. --S. L. Sullivan, Catawba College

List of Figures
xv
Preface xvii
1 Introduction
1(26)
1.1 Some Combinatorial Examples
1(12)
1.2 Sets, Relations, and Proof Techniques
13(2)
1.3 Two Principles of Enumeration
15(3)
1.4 Graphs
18(2)
1.5 Systems of Distinct Representatives
20(7)
Exercises 1A
22(1)
Exercises 1B
23(2)
Problems
25(2)
2 Fundamentals of Enumeration
27(18)
2.1 Permutations and Combinations
27(2)
2.2 Applications of P(n, k) And
29(3)
2.3 Permutations and Combinations of Multisets
32(4)
2.4 Applications and Subtle Errors
36(2)
2.5 Algorithms
38(7)
Exercises 2A
39(2)
Exercises 2B
41(1)
Problems
42(3)
3 Probability
45(20)
3.1 Introduction
45(1)
3.2 Some Definitions and Easy Examples
45(3)
3.3 Events and Probabilities
48(4)
3.4 Three Interesting Examples
52(2)
3.5 Probability Models
54(1)
3.6 Bernoulli Trials
55(1)
3.7 The Probabilities in Poker
56(1)
3.8 The Wild Card Poker Paradox
57(8)
Exercises 3A
58(2)
Exercises 3B
60(2)
Problems
62(3)
4 The Pigeonhole Principle and Ramsey's Theorem
65(22)
4.1 The Pigeonhole Principle
65(1)
4.2 Applications of the Pigeonhole Principle
66(3)
4.3 Ramsey's Theorem---The Graphical Case
69(3)
4.4 Ramsey Multiplicity
72(2)
4.5 Sum-Free Sets
74(3)
4.6 Bounds on Ramsey Numbers
77(4)
4.7 The General Form of Ramsey's Theorem
81(6)
Exercises 4A
81(1)
Exercises 4B
82(2)
Problems
84(3)
5 The Principle of Inclusion and Exclusion
87(16)
5.1 Unions of Events
87(2)
5.2 The Principle
89(4)
5.3 Combinations with Limited Repetitions
93(2)
5.4 Derangements
95(8)
Exercises 5A
98(1)
Exercises 5B
99(2)
Problems
101(2)
6 Generating Functions and Recurrence Relations
103(18)
6.1 Generating Functions
103(4)
6.2 Recurrence Relations
107(5)
6.3 From Generating Function to Recurrence
112(1)
6.4 Exponential Generating Functions
113(8)
Exercises 6A
115(2)
Exercises 6B
117(1)
Problems
118(3)
7 Catalan, Bell, and Stirling Numbers
121(22)
7.1 Introduction
121(1)
7.2 Catalan Numbers
122(4)
7.3 Stirling Numbers of the Second Kind
126(5)
7.4 Bell Numbers
131(1)
7.5 Stirling Numbers of the First Kind
132(3)
7.6 Computer Algebra and Other Electronic Systems
135(8)
Exercises 7A
137(2)
Exercises 7B
139(1)
Problems
140(3)
8 Symmetries and the Polya-Redfield Method
143(20)
8.1 Introduction
143(1)
8.2 Basics of Groups
144(6)
8.3 Permutations and Colorings
150(1)
8.4 An Important Counting Theorem
151(3)
8.5 The Polya and Redfield Theorem
154(9)
Exercises 8A
158(1)
Exercises 8B
159(1)
Problems
160(3)
9 Partially Ordered Sets
163(14)
9.1 Introduction
163(1)
9.2 Examples and Definitions
164(2)
9.3 Bounds and Lattices
166(2)
9.4 Isomorphism and Cartesian Products
168(3)
9.5 Extremal Set Theory: Sperner's and Dilworth's Theorems
171(6)
Exercises 9A
175(1)
Exercises 9B
176(1)
Problems
176(1)
10 Introduction to Graph Theory
177(16)
10.1 Degrees
177(3)
10.2 Paths and Cycles in Graphs
180(3)
10.3 Maps and Graph Coloring
183(10)
Exercises 10A
187(2)
Exercises 10B
189(1)
Problems
190(3)
11 Further Graph Theory
193(30)
11.1 Euler Walks and Circuits
193(5)
11.2 Application of Euler Circuits to Mazes
198(2)
11.3 Hamilton Cycles
200(4)
11.4 Trees
204(4)
11.5 Spanning Trees
208(15)
Exercises 11A
214(4)
Exercises 11B
218(3)
Problems
221(2)
12 Coding Theory
223(18)
12.1 Errors and Noise
223(1)
12.2 The Venn Diagram Code
224(2)
12.3 Binary Codes, Weight, and Distance
226(3)
12.4 Linear Codes
229(3)
12.5 Hamming Codes
232(2)
12.6 Codes and the Hat Problem
234(1)
12.7 Variable-Length Codes and Data Compression
235(6)
Exercises 12A
237(1)
Exercises 12B
238(1)
Problems
239(2)
13 Latin Squares
241(24)
13.1 Introduction
241(4)
13.2 Orthogonality
245(6)
13.3 Idempotent Latin Squares
251(2)
13.4 Partial Latin Squares and Subsquares
253(2)
13.5 Applications
255(10)
Exercises 13A
259(3)
Exercises 13B
262(1)
Problems
263(2)
14 Balanced Incomplete Block Designs
265(20)
14.1 Design Parameters
266(4)
14.2 Fisher's Inequality
270(2)
14.3 Symmetric Balanced Incomplete Block Designs
272(2)
14.4 New Designs from Old
274(2)
14.5 Difference Methods
276(9)
Exercises 14A
279(2)
Exercises 14B
281(1)
Problems
282(3)
15 Linear Algebra Methods in Combinatorics
285(20)
15.1 Recurrences Revisited
285(2)
15.2 State Graphs and the Transfer Matrix Method
287(7)
15.3 Kasteleyn's Permanent Method
294(11)
Exercises 15A
299(2)
Exercises 15B
301(1)
Problems
302(3)
Appendix A Sets; Proof Techniques
305(20)
AA.1 Sets and Basic Set Operations
305(8)
AA.2 The Principle of Mathematical Induction
313(2)
AA.3 Some Applications of Induction
315(2)
AA.4 Binary Relations on Sets
317(8)
Exercises A
319(1)
Exercises B
320(5)
Appendix B Matrices and Vectors
325(14)
B.1 Definitions
325(3)
B.2 Vector and Matrix Products
328(2)
B.3 Inverses
330(2)
B.4 Determinants
332(7)
Exercises A
333(1)
Exercises B
334(5)
Appendix C Some Combinatorial People
339(8)
Solutions to Set A Exercises 347(30)
Hints for Problems 377(6)
Solutions to Problems 383(24)
References 407(10)
Index 417
W.D. Wallis is Professor Emeritus of Southern Illiniois University. John C George is Asscoiate Professor at Gordon State College.
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