Chapter 5Partitions and Permutations

5.1 Stirling Subset Numbers5.2 Stirling Cycle Numbers5.3 Inversions and Ascents5.4 Derangements5.5 Exponential Generating Functions5.6 Posets and Lattices

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2 Chapter 5 Partitions and Permutations

Section 5.1 Stirling Subset Numbers 3

5.1 STIRLING SUBSET NUMBERS

Non-Distinctness of Cells of a Partition

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Section 5.1 Stirling Subset Numbers 5

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Every Cell of a Partition is Non-Empty

Section 5.1 Stirling Subset Numbers 7

Distinctness of Objects

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The Type of a Partition

Section 5.1 Stirling Subset Numbers 9

Stirling’s Subset Number Recurrence

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Stirling’s Triangle for Subset Numbers

Table 5.1.1

Section 5.1 Stirling Subset Numbers 11

Rows Are Log-Concave

Fig 5.1.1

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Section 5.1 Stirling Subset Numbers 13

Bell Numbers

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Section 5.1 Stirling Subset Numbers 15

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Column-Sum Formulas

Section 5.1 Stirling Subset Numbers 17

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Section 5.1 Stirling Subset Numbers 19

Southeast Diagonal Sum

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Stirling Numbers of the Second Kind

Section 5.1 Stirling Subset Numbers 21

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Section 5.1 Stirling Subset Numbers 23

Table 5.1.2

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Section 5.2 Stirling Cycle Numbers 25

5.2 STIRLING CYCLE NUMBERS

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Section 5.2 Stirling Cycle Numbers 27

Non-Distinctness of the Cycles

Stirling’s Cycle Number Recurrence

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Stirling’s Triangle for Cycle Numbers

Section 5.2 Stirling Cycle Numbers 29

Table 5.2.1

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Section 5.2 Stirling Cycle Numbers 31

Rows are Log-Concave

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Section 5.2 Stirling Cycle Numbers 33

Fig 5.2.1

Row Sums

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Section 5.2 Stirling Cycle Numbers 35

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Section 5.2 Stirling Cycle Numbers 37

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Section 5.2 Stirling Cycle Numbers 39

Columns

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Section 5.2 Stirling Cycle Numbers 41

Southeast Diagonal

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Stirling Numbers of the First Kind

Section 5.2 Stirling Cycle Numbers 43

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Section 5.2 Stirling Cycle Numbers 45

Table 5.2.2

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Section 5.3 Inversions and Ascents 47

5.3 INVERSIONS AND ASCENTS

Inversions

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Section 5.3 Inversions and Ascents 49

Table 5.3.1

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Section 5.3 Inversions and Ascents 51

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Ascents

Section 5.3 Inversions and Ascents 53

Eulerian Numbers

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Table 5.3.2

Section 5.3 Inversions and Ascents 55

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5.4 DERANGEMENTS

Section 5.4 Derangements 57

Table 5.4.1

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Section 5.4 Derangements 59

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5.5 EXPONENTIAL GEN FUNCTIONS

Section 5.5 Exponential Gen Functions 61

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Section 5.5 Exponential Gen Functions 63

Counting Ordered Selections

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Section 5.5 Exponential Gen Functions 65

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Counting Certain Kinds of Strings

Section 5.5 Exponential Gen Functions 67

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Section 5.5 Exponential Gen Functions 69

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An Application To Stirling Subset #s

Section 5.5 Exponential Gen Functions 71

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An EGF for Derangement Numbers

Section 5.5 Exponential Gen Functions 73

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Section 5.5 Exponential Gen Functions 75

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5.6 POSETS AND LATTICES

Section 5.6 Posets and Lattices 77

Products of Sets

Cover Digraph

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Fig 5.6.1

The Boolean Poset

Section 5.6 Posets and Lattices 79

Fig 5.6.2

The Divisibility Poset

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Fig 5.6.3

The Partition Poset

Section 5.6 Posets and Lattices 81

Fig 5.6.4

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Inversion-Dominance Ordering on Perms

Section 5.6 Posets and Lattices 83

Fig 5.6.5

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Minimal and Maximal Elements

Fig 5.6.6

Section 5.6 Posets and Lattices 85

Lattice Property

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Section 5.6 Posets and Lattices 87

Fig 5.6.7

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Fig 5.6.8

Poset Isomorphism

Section 5.6 Posets and Lattices 89

Fig 5.6.9

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Fig 5.6.10

Chains and Antichains

Section 5.6 Posets and Lattices 91

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Section 5.6 Posets and Lattices 93

Fig 5.6.11

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Ranked Posets

Section 5.6 Posets and Lattices 95

Fig 5.6.12

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Linear Extensions

Section 5.6 Posets and Lattices 97

Algorithm 5.6.1:

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Dilworth’s Theorem

Section 5.6 Posets and Lattices 99

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Section 5.6 Posets and Lattices 101

Fig 5.6.13

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