The theory of integer partitions is a subject of enduring interest. A major research area in its own right, it has found numerous applications, and celebrated results such as the Rogers-Ramanujan identities make it a topic filled with the true romance of mathematics. The aim in this introductory textbook is to provide an accessible and wide ranging introduction to partitions, without requiring anything more of the reader than some familiarity with polynomials and infinite series. Many exercises are included, together with some solutions and helpful hints. The book has a short introduction followed by an initial chapter introducing Euler's famous theorem on partitions with odd parts and partitions with distinct parts. This is followed by chapters titled: Ferrers Graphs, The Rogers-Ramanujan Identities, Generating Functions, Formulas for Partition Functions, Gaussian Polynomials, Durfee Squares, Euler Refined, Plane Partitions, Growing Ferrers Boards, and Musings.
Author(s): Andrews G.E., Eriksson K.
Publisher: CUP
Year: 2004
Language: English
Pages: 154
Front cover......Page 1
Abstract......Page 2
Dedication......Page 3
Title page......Page 5
Date-line......Page 6
Contents......Page 7
Preface......Page 11
1 Introduction......Page 13
2.1 Set terminology......Page 17
2.2 Bijective proofs of partition identities......Page 18
2.3 A bijection for Euler's identity......Page 20
2.4 Euler pairs......Page 22
3 Ferrers graphs......Page 26
3.1 Ferrers graphs and Ferrers boards......Page 27
3.2 Conj ugate partitions......Page 28
3.3 An upper bound on $p(n)$......Page 31
3.4 Bressoud's beautiful bijection......Page 35
3.5 Euler's pentagonal number theorem......Page 36
4.1 A fundamental type of partition identity......Page 41
4.2 Discovering the first Rogers-Ramanujan identity......Page 43
4.3 Alder's conjecture......Page 45
4.4 Schur's theorem......Page 47
4.5 Looking for a bijective proof of the first Rogers-Ramanujan identity......Page 51
4.6 The impact of the Rogers-Ramanujan identities......Page 53
5.1 Generating functions as products......Page 54
5.2 Euler's theorem......Page 59
5.3 Two variable-generating functions......Page 60
5.4 Euler's pentagonal number theorem......Page 61
5.5 Congruences for $p(n)$......Page 63
5.6 Rogers-Ramanujan revisited......Page 64
6.1 Formula for $p(n,1)$ and $p(n,2)$......Page 67
6.2 A formula for $p(n,3)$......Page 69
6.3 A formula for $p(n,4)$......Page 70
6.4 $\Lim_{n\to\infty} p(n)^{1/n} = 1$......Page 73
7.1 Properties of the binomial numbers......Page 76
7.2 Lattice paths and the $q$-binomial numbers......Page 79
7.3 The $q$-binomial theorem and the $q$-binomial series......Page 81
7.4 Gaussian polynomial identities......Page 83
7.5 Limiting values of Gaussian polynomials......Page 86
8.1 Durfee squares and generating functions......Page 87
8.2 Frobenius symbols......Page 90
8.3 Jacobi's triple product identity......Page 91
8.4 The Rogers-Ramanuj an identities......Page 93
8.5 Successive Durfee squares......Page 97
9.1 Sylvester's refinement of Euler......Page 100
9.2 Fine's refinement......Page 102
9.3 Lecture hall partitions......Page 104
10.1 Ferrers graphs and rhombus tilings......Page 111
10.2 MacMahon's formulas......Page 113
10.3 The formula for $\pi_r(h,j_i;q)$......Page 115
11.1 Random partitions......Page 118
11.2 Posets of partitions......Page 119
11.3 The hook length formula......Page 122
11.4 Randomly growing Ferrers boards......Page 125
11.5 Domino tilings......Page 127
11.6 The arctic circle theorem......Page 128
12.1 What have we left out?......Page 133
12.2 Where can you go to undertake new explorations?......Page 135
12.3 Where can one study the history of partitions?......Page 136
12.4 Are there any unsolved problems left?......Page 137
A On the convergence of infinite series and products......Page 138
B References......Page 141
C Solutions and hints to selected exercises......Page 144
Index......Page 151
Back cover......Page 154
