The Theory of PartitionsCambridge University Press, 28.07.1998 - 255 Seiten This book develops the theory of partitions. Simply put, the partitions of a number are the ways of writing that number as sums of positive integers. For example, the five partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. Surprisingly, such a simple matter requires some deep mathematics for its study. This book considers the many theoretical aspects of this subject, which have in turn recently found applications to statistical mechanics, computer science and other branches of mathematics. With minimal prerequisites, this book is suitable for students as well as researchers in combinatorics, analysis, and number theory. |
Inhalt
The Elementary Theory of Partitions | 1 |
12 Infinite Product Generating Functions of One Variable | 3 |
13 Graphical Representation of Partitions | 6 |
Examples | 13 |
Notes | 14 |
Infinite Series Generating Functions | 16 |
22 Elementary SeriesProduct Identities | 17 |
23 Applications to Partitions | 23 |
Notes | 137 |
References | 138 |
Sieve Methods Related to Partitions | 139 |
93 A Sieve for Successive Ranks | 142 |
Examples | 156 |
Notes | 158 |
Congruence Properties of Partition Functions | 159 |
102 Rödseths Theorem for Binary Partitions | 161 |
Examples | 28 |
Notes | 30 |
References | 31 |
Restricted Partitions and Permutations | 33 |
33 Properties of Gaussian Polynomials | 35 |
34 Permutations and Gaussian Multinomial Coefficients | 39 |
35 The Unimodal Property | 45 |
Examples | 49 |
Notes | 51 |
References | 52 |
Compositions and Simon Newcombs Problem | 54 |
43 Vector Compositions | 57 |
44 Simon Newcombs Problem | 59 |
Examples | 63 |
Notes | 65 |
The HardyRamanujanRademacher Expansion of pn | 68 |
52 The Formula for pn | 71 |
Examples | 81 |
Notes | 85 |
References | 86 |
The Asymptotics of Infinite Product Generating Functions | 88 |
62 Proof of Theorem 62 | 89 |
63 Applications of Theorem 62 | 97 |
Notes | 100 |
| 101 | |
Identities of the RogersRamanujan Type | 103 |
72 The Generating Functions | 106 |
73 The RogersRamanujan Identities and Gordons Generalization | 109 |
74 The GollnitzGordon Identities and Their Generalization | 113 |
Examples | 116 |
Notes | 118 |
81 Introduction | 124 |
84 Linked Partition Ideals | 128 |
Examples | 136 |
103 Ramanujans Conjecture for 5 | 167 |
Examples | 175 |
Notes | 177 |
References | 178 |
HigherDimensional Partitions | 179 |
113 The KnuthSchensted Correspondence | 184 |
114 HigherDimensional Partitions | 189 |
Examples | 198 |
Notes | 199 |
References | 200 |
Vector or Multipartite Partitions | 202 |
123 Bell Polynomials and Formulas for Multipartite Partition Functions | 204 |
124 Restricted Bipartite Partitions | 207 |
Examples | 209 |
Notes | 210 |
| 211 | |
Partitions in Combinatorics | 212 |
133 Partitions of Sets | 214 |
134 The Combinatorics of Symmetric Functions | 221 |
Examples | 225 |
Notes | 227 |
References | 228 |
Computations for Partitions | 230 |
143 Algorithms from Generating Functions | 233 |
144 Computations for HigherDimensional Partitions | 234 |
145 Brief Tables of Partition Functions | 237 |
147 Table of Gaussian Polynomials | 240 |
148 Guide to Tables | 243 |
| 244 | |
| 245 | |
| 250 | |
| 252 | |
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Häufige Begriffe und Wortgruppen
1976 by Addison-Wesley a₁ a₂ Amer analytic Andrews Applications basic hypergeometric series Bell polynomials C₁ C₂ Carlitz Chapter Combinatorial Theory comparing coefficients compositions computation congruences conjecture Corollary 1.2 correspondence define DEFINITION denote the number Durfee square enumerated equals the number Euler exactly Example Ferrers graph finite follows Gaussian polynomials Gian-Carlo Rota graphical representation Hence higher-dimensional partitions i)-positive ideals of order infinite product inv(m Lemma LeVeque linked partition ideal London Math m₁ m₂ MacMahon modular modulus multipartite partition multiset n₁ n₂ number of partitions Number Theory ordinary partition partition function partition theorem pentagonal number theorem permutations plane partitions positive integers problems Proc proof of Theorem prove q-analog Ramanujan reprinted by Chelsea result Rogers-Ramanujan identities Section sequence Sylvester symmetric Theorem 1.1 theory of partitions vector πο Σ Σ л₁
Verweise auf dieses Buch
Univariate Discrete Distributions Norman L. Johnson,Adrienne W. Kemp,Samuel Kotz Eingeschränkte Leseprobe - 2005 |