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

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Section 3.1 Normalizing Summations 3

3.1 NORMALIZING SUMMATIONS

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Section 3.1 Normalizing Summations 5

Sums over Sets

Fig 3.1.1

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Section 3.1 Normalizing Summations 7

Iverson Truth Function

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Algebraic Regrouping

Section 3.1 Normalizing Summations 9

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Section 3.1 Normalizing Summations 11

Harmonic Numbers

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Fig 3.1.2

Section 3.1 Normalizing Summations 13

GKP Notations

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3.2 PERTURBATION

Section 3.2 Perturbation 15

A Classical Example of Perturbation

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Section 3.2 Perturbation 17

Indirect Perturbation

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Section 3.2 Perturbation 19

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Section 3.3 Summing with OGF’s 21

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3.3 SUMMING WITH OGF’S

Revisiting Examples

Section 3.3 Summing with OGF’s 23

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Section 3.3 Summing with OGF’s 25

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Section 3.3 Summing with OGF’s 27

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3.4 FINITE CALCULUS

Section 3.4 Finite Calculus 29

Summing a Polynomial

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Section 3.4 Finite Calculus 31

Formula for Summing Exponentials

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Section 3.4 Finite Calculus 33

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Falling Negative Powers

Section 3.4 Finite Calculus 35

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Section 3.4 Finite Calculus 37

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Harmonic Numbers

Section 3.4 Finite Calculus 39

Product Formula

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Summation by Parts

Section 3.4 Finite Calculus 41

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Table 3.4.1

Section 3.4 Finite Calculus 43

Table 3.4.2

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3.5 ITERATED SUMS

Section 3.5 Iterated Sums 45

Independent Indices

Fig 3.5.1

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Section 3.5 Iterated Sums 47

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Dependent Indices

Fig 3.5.2

Section 3.5 Iterated Sums 49

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Section 3.5 Iterated Sums 51

Linear Partitioning: Floor Sums

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Section 3.5 Iterated Sums 53

Table 3.5.1

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Section 3.5 Iterated Sums 55

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Section 3.6 Inclusion-Exclusion 57

3.6 INCLUSION-EXCLUSION

Venn Diagrams for Overlapping Subsets

Fig 3.6.1

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Section 3.6 Inclusion-Exclusion 59

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Venn Diagrams for 3 or More Subsets

Section 3.6 Inclusion-Exclusion 61

Fig 3.6.2

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Fig 3.6.3

Section 3.6 Inclusion-Exclusion 63

Context for Inclusion-Exclusion

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Formulas for Inclusion-Exclusion

Section 3.6 Inclusion-Exclusion 65

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Section 3.6 Inclusion-Exclusion 67

Stirling Subset Numbers

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Section 3.6 Inclusion-Exclusion 69

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Section 3.6 Inclusion-Exclusion 71

Derangements

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Section 3.6 Inclusion-Exclusion 73

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Counting Bipartite Matchings

Section 3.6 Inclusion-Exclusion 75

Fig 3.6.4

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Section 3.6 Inclusion-Exclusion 77

Fig 3.6.5

Chromatic Polynomials

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Fig 3.6.6

Section 3.6 Inclusion-Exclusion 79

Fig 3.6.7

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Section 3.6 Inclusion-Exclusion 81

Fig 3.6.8

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Section 3.6 Inclusion-Exclusion 83