Chapter 3Evaluating Sums
3.1 Normalizing Summations3.2 Perturbation3.3 Summing with Generating Functions3.4 Finite Calculus3.5 Iteration and Partitioning of Sums3.6 Inclusion-Exclusion
1
2 Chapter 3 Evaluating Sums
Section 3.1 Normalizing Summations 3
3.1 NORMALIZING SUMMATIONS
4 Chapter 3 Evaluating Sums
Section 3.1 Normalizing Summations 5
Sums over Sets
Fig 3.1.1
6 Chapter 3 Evaluating Sums
Section 3.1 Normalizing Summations 7
Iverson Truth Function
8 Chapter 3 Evaluating Sums
Algebraic Regrouping
Section 3.1 Normalizing Summations 9
10 Chapter 3 Evaluating Sums
Section 3.1 Normalizing Summations 11
Harmonic Numbers
12 Chapter 3 Evaluating Sums
Fig 3.1.2
Section 3.1 Normalizing Summations 13
GKP Notations
14 Chapter 3 Evaluating Sums
3.2 PERTURBATION
Section 3.2 Perturbation 15
A Classical Example of Perturbation
16 Chapter 3 Evaluating Sums
Section 3.2 Perturbation 17
Indirect Perturbation
18 Chapter 3 Evaluating Sums
Section 3.2 Perturbation 19
20 Chapter 3 Evaluating Sums
Section 3.3 Summing with OGF’s 21
22 Chapter 3 Evaluating Sums
3.3 SUMMING WITH OGF’S
Revisiting Examples
Section 3.3 Summing with OGF’s 23
24 Chapter 3 Evaluating Sums
Section 3.3 Summing with OGF’s 25
26 Chapter 3 Evaluating Sums
Section 3.3 Summing with OGF’s 27
28 Chapter 3 Evaluating Sums
3.4 FINITE CALCULUS
Section 3.4 Finite Calculus 29
Summing a Polynomial
30 Chapter 3 Evaluating Sums
Section 3.4 Finite Calculus 31
Formula for Summing Exponentials
32 Chapter 3 Evaluating Sums
Section 3.4 Finite Calculus 33
34 Chapter 3 Evaluating Sums
Falling Negative Powers
Section 3.4 Finite Calculus 35
36 Chapter 3 Evaluating Sums
Section 3.4 Finite Calculus 37
38 Chapter 3 Evaluating Sums
Harmonic Numbers
Section 3.4 Finite Calculus 39
Product Formula
40 Chapter 3 Evaluating Sums
Summation by Parts
Section 3.4 Finite Calculus 41
42 Chapter 3 Evaluating Sums
Table 3.4.1
Section 3.4 Finite Calculus 43
Table 3.4.2
44 Chapter 3 Evaluating Sums
3.5 ITERATED SUMS
Section 3.5 Iterated Sums 45
Independent Indices
Fig 3.5.1
46 Chapter 3 Evaluating Sums
Section 3.5 Iterated Sums 47
48 Chapter 3 Evaluating Sums
Dependent Indices
Fig 3.5.2
Section 3.5 Iterated Sums 49
50 Chapter 3 Evaluating Sums
Section 3.5 Iterated Sums 51
Linear Partitioning: Floor Sums
52 Chapter 3 Evaluating Sums
Section 3.5 Iterated Sums 53
Table 3.5.1
54 Chapter 3 Evaluating Sums
Section 3.5 Iterated Sums 55
56 Chapter 3 Evaluating Sums
Section 3.6 Inclusion-Exclusion 57
3.6 INCLUSION-EXCLUSION
Venn Diagrams for Overlapping Subsets
Fig 3.6.1
58 Chapter 3 Evaluating Sums
Section 3.6 Inclusion-Exclusion 59
60 Chapter 3 Evaluating Sums
Venn Diagrams for 3 or More Subsets
Section 3.6 Inclusion-Exclusion 61
Fig 3.6.2
62 Chapter 3 Evaluating Sums
Fig 3.6.3
Section 3.6 Inclusion-Exclusion 63
Context for Inclusion-Exclusion
64 Chapter 3 Evaluating Sums
Formulas for Inclusion-Exclusion
Section 3.6 Inclusion-Exclusion 65
66 Chapter 3 Evaluating Sums
Section 3.6 Inclusion-Exclusion 67
Stirling Subset Numbers
68 Chapter 3 Evaluating Sums
Section 3.6 Inclusion-Exclusion 69
70 Chapter 3 Evaluating Sums
Section 3.6 Inclusion-Exclusion 71
Derangements
72 Chapter 3 Evaluating Sums
Section 3.6 Inclusion-Exclusion 73
74 Chapter 3 Evaluating Sums
Counting Bipartite Matchings
Section 3.6 Inclusion-Exclusion 75
Fig 3.6.4
76 Chapter 3 Evaluating Sums
Section 3.6 Inclusion-Exclusion 77
Fig 3.6.5
Chromatic Polynomials
78 Chapter 3 Evaluating Sums
Fig 3.6.6
Section 3.6 Inclusion-Exclusion 79
Fig 3.6.7
80 Chapter 3 Evaluating Sums
Section 3.6 Inclusion-Exclusion 81
Fig 3.6.8
82 Chapter 3 Evaluating Sums
Section 3.6 Inclusion-Exclusion 83