Chapter 5 - Partitions and Permutations

Chapter

5

Partitions and Permutations 5.1 Stirling Subset Numbers 5.2 Stirling Cycle Numbers 5.3 Inversions and Ascents 5.4 Derangements 5.5 Exponential Generating Functions 5.6 Posets and Lattices

1

2

Chapter 5 Partitions and Permutations

Section 5.1 Stirling Subset Numbers

5.1 STIRLING SUBSET NUMBERS

Non-Distinctness of Cells of a Partition

3

4

Chapter 5 Partitions and Permutations

Section 5.1 Stirling Subset Numbers

5

6

Chapter 5 Partitions and Permutations

Every Cell of a Partition is Non-Empty

Section 5.1 Stirling Subset Numbers

Distinctness of Objects

7

8

Chapter 5 Partitions and Permutations

The Type of a Partition

Section 5.1 Stirling Subset Numbers

Stirling’s Subset Number Recurrence

9

10

Chapter 5 Partitions and Permutations

Stirling’s Triangle for Subset Numbers

Table 5.1.1

Section 5.1 Stirling Subset Numbers

Rows Are Log-Concave

Fig 5.1.1

11

12

Chapter 5 Partitions and Permutations

Section 5.1 Stirling Subset Numbers

Bell Numbers

13

14

Chapter 5 Partitions and Permutations

Section 5.1 Stirling Subset Numbers

15

16

Chapter 5 Partitions and Permutations

Column-Sum Formulas

Section 5.1 Stirling Subset Numbers

17

18

Chapter 5 Partitions and Permutations

Section 5.1 Stirling Subset Numbers

Southeast Diagonal Sum

19

20

Chapter 5 Partitions and Permutations

Stirling Numbers of the Second Kind

Section 5.1 Stirling Subset Numbers

21

22

Chapter 5 Partitions and Permutations

Section 5.1 Stirling Subset Numbers

Table 5.1.2

23

24

Chapter 5 Partitions and Permutations

Section 5.2 Stirling Cycle Numbers

5.2 STIRLING CYCLE NUMBERS

25

26

Chapter 5 Partitions and Permutations

Section 5.2 Stirling Cycle Numbers

Non-Distinctness of the Cycles

Stirling’s Cycle Number Recurrence

27

28

Chapter 5 Partitions and Permutations

Stirling’s Triangle for Cycle Numbers

Section 5.2 Stirling Cycle Numbers

Table 5.2.1

29

30

Chapter 5 Partitions and Permutations

Section 5.2 Stirling Cycle Numbers

Rows are Log-Concave

31

32

Chapter 5 Partitions and Permutations

Section 5.2 Stirling Cycle Numbers

Fig 5.2.1

Row Sums

33

34

Chapter 5 Partitions and Permutations

Section 5.2 Stirling Cycle Numbers

35

36

Chapter 5 Partitions and Permutations

Section 5.2 Stirling Cycle Numbers

37

38

Chapter 5 Partitions and Permutations

Section 5.2 Stirling Cycle Numbers

Columns

39

40

Chapter 5 Partitions and Permutations

Section 5.2 Stirling Cycle Numbers

Southeast Diagonal

41

42

Chapter 5 Partitions and Permutations

Stirling Numbers of the First Kind

Section 5.2 Stirling Cycle Numbers

43

44

Chapter 5 Partitions and Permutations

Section 5.2 Stirling Cycle Numbers

Table 5.2.2

45

46

Chapter 5 Partitions and Permutations

Section 5.3 Inversions and Ascents

5.3 INVERSIONS AND ASCENTS

Inversions

47

48

Chapter 5 Partitions and Permutations

Section 5.3 Inversions and Ascents

Table 5.3.1

49

50

Chapter 5 Partitions and Permutations

Section 5.3 Inversions and Ascents

51

52

Ascents

Chapter 5 Partitions and Permutations

Section 5.3 Inversions and Ascents

Eulerian Numbers

53

54

Table 5.3.2

Chapter 5 Partitions and Permutations

Section 5.3 Inversions and Ascents

55

56

Chapter 5 Partitions and Permutations

5.4 DERANGEMENTS

Section 5.4 Derangements

Table 5.4.1

57

58

Chapter 5 Partitions and Permutations

Section 5.4 Derangements

59

60

Chapter 5 Partitions and Permutations

5.5 EXPONENTIAL GEN FUNCTIONS

Section 5.5 Exponential Gen Functions

61

62

Chapter 5 Partitions and Permutations

Section 5.5 Exponential Gen Functions

Counting Ordered Selections

63

64

Chapter 5 Partitions and Permutations

Section 5.5 Exponential Gen Functions

65

66

Chapter 5 Partitions and Permutations

Counting Certain Kinds of Strings

Section 5.5 Exponential Gen Functions

67

68

Chapter 5 Partitions and Permutations

Section 5.5 Exponential Gen Functions

69

70

Chapter 5 Partitions and Permutations

An Application To Stirling Subset #s

Section 5.5 Exponential Gen Functions

71

72

Chapter 5 Partitions and Permutations

An EGF for Derangement Numbers

Section 5.5 Exponential Gen Functions

73

74

Chapter 5 Partitions and Permutations

Section 5.5 Exponential Gen Functions

75

76

Chapter 5 Partitions and Permutations

5.6 POSETS AND LATTICES

Section 5.6 Posets and Lattices

Products of Sets

Cover Digraph

77

78

Chapter 5 Partitions and Permutations

Fig 5.6.1

The Boolean Poset

Section 5.6 Posets and Lattices

Fig 5.6.2

The Divisibility Poset

79

80

Chapter 5 Partitions and Permutations

Fig 5.6.3

The Partition Poset

Section 5.6 Posets and Lattices

Fig 5.6.4

81

82

Chapter 5 Partitions and Permutations

Inversion-Dominance Ordering on Perms

Section 5.6 Posets and Lattices

Fig 5.6.5

83

84

Chapter 5 Partitions and Permutations

Minimal and Maximal Elements

Fig 5.6.6

Section 5.6 Posets and Lattices

Lattice Property

85

86

Chapter 5 Partitions and Permutations

Section 5.6 Posets and Lattices

Fig 5.6.7

87

88

Chapter 5 Partitions and Permutations

Fig 5.6.8

Poset Isomorphism

Section 5.6 Posets and Lattices

Fig 5.6.9

89

90

Chapter 5 Partitions and Permutations

Fig 5.6.10

Chains and Antichains

Section 5.6 Posets and Lattices

91

92

Chapter 5 Partitions and Permutations

Section 5.6 Posets and Lattices

Fig 5.6.11

93

94

Ranked Posets

Chapter 5 Partitions and Permutations

Section 5.6 Posets and Lattices

Fig 5.6.12

95

96

Chapter 5 Partitions and Permutations

Linear Extensions

Section 5.6 Posets and Lattices

Algorithm 5.6.1:

97

98

Chapter 5 Partitions and Permutations

Dilworth’s Theorem

Section 5.6 Posets and Lattices

99

100

Chapter 5 Partitions and Permutations

Section 5.6 Posets and Lattices

Fig 5.6.13

101

102

Chapter 5 Partitions and Permutations