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