Principles of Combinatorics (eBook)

Front Cover 1
Principles of Combinatorics 4
Copyright Page 5
Contents 6
Foreword 8
What Is Combinatorics? 12
First Aspect: Study of a Known Configuration 14
Second Aspect : Investigation of an Unknown Configuration 14
Third Aspect : Counting Configurations 16
Fourth Aspect : Approximate Counting of Configurations 18
Fifth Aspect : Enumeration of Configurations 19
Sixth Aspect: Optimization 20
References 21
Chapter 1. The Elemientary Counting Functions 24
1. Mappings of Finite Sets 24
2. The Cardinality of the Cartesian Product A x X 27
3. Number of Subsets of a Finite Set A 28
4. Numbers mn or Mappings of X into A 30
5. Numbers [M]n, or Injections of X into A 30
6. Numbers [M]n 33
7. Numbers [m]n/n!, or Increasing Mappings of X Into A 34
8. Binomial Numbers 36
9. Multinomial Numbers(n n1,n2,....np) 43
10. Stirling Numbers Snm, or Partitions of n Objects into m Classes 48
11. Bell Exponential Number Bn, or the Number of Partitions of n Objects 53
References 55
Chapter 2. Partition Problems 58
1. Pnm, or the Number of Partitions of Integer n into m Parts 58
2. Pn,h, or the Number of Partitions of the Integer n Having h as the Smallest Part 67
3. Counting the Standard Tableaus Associated with a Partition of n 70
4. Standard Tableaus and Young's Lattice 80
References 82
Chapter 3. Inversion Formulas and Their Applications 84
1. Differential Operator Associated with a Family of Polynomials 84
2. The Möbius Function 91
3. Sieve Formulas 99
4. Distributions 106
5. Counting Trees 110
References 120
Chapter 4. Permutation Groups 122
1. Introduction 122
2. Cycles of a Permutation 130
3. Orbits of a Permutation Group 134
4. Parity of a Permutation 137
5. Decomposition Problems 150
References 158
Chapter 5. Pólya's Theorem 160
1. Counting Schemata Relative to a Group of Permutations of Objects 160
2. Counting Schemata Relative to an Arbitrary Group 167
3. A Theorem of de Bruijn 175
4. Computing the Cycle Index 181
References 184
Index 186