内容简介
Integer partition is one of the most fundamental research subjectsin combinatorics. The theory of partition has attracted the attention of many famous mathematicians and developed for centuries.
This is a book about integer partition identities. We startfrom some basic concepts in the theory of partition. Then we focus on two family of partition identities after Euler’spartition theorem. One family of identities involve partitions with restrictions on the differences of consecutive parts. Rogers- Ramanujan identities are the most important identities in this family. We present some of the most famous results: identities of Rogers-Ramanujan type, Schur’s theorem, G¨ollnitz-Gordon theorem as well as some overpartition analogues. The otherfamily of partition identities are about partitions with restrictions on the quotient of consecutive parts. We present some quite recent results involving lecture hall partitions, anti-lecture hall compositions, a-lecture hall partitions and truncated lecture hall partitions.
Over the years I have been assisted greatly by many persons and institutions. Among them, I wish to acknowledge the School of Mathematics in Dongbei University of Finance and Economics,the Center for Combinatorics in Nankai University and the National Science Foundation (Project No. 11501089). I am deeply indebted to my Ph.D. supervisor Professor Yongchuan Chen, who leads me into the fields of combinatorics and integer 2 Integer Partitions with Difference Conditions and Quotient Conditions and Related q-series Identities
partitions. I would like to show great appreciations to my wonderful research partner Professor Yahui Shi, without whose joint efforts I could not obtain the results in partition theory.
章节目录
Chapter 1 Introduction 1
Chapter 2 Basic Concepts 5
2.1 Interger partitions and compositions . . . . . . . . . . 5
2.2 Ferrers graphs . . . . . . . . . . . . . . . . . . . . . 6
2.3 Generating functions . . . . . . . . . . . . . . . . . . 9
2.4 Overpartitions . . . . . . . . . . . . . . . . . . . . . 15
Chapter 3 Partition identities with difference conditions 19
3.1 Euler’s partition identity . . . . . . . . . . . . . . . . 19
3.2 Rogers-Ramanujan identities . . . . . . . . . . . . . . 22
3.3 Schur’s theorem and G¨ollnitz-Gordon theorem . . . . . . 31
3.4 Overpartition analogues . . . . . . . . . . . . . . . . 33
Chapter 4 Partition identities with quotient conditions 55
4.1 Lecture hall theorem . . . . . . . . . . . . . . . . . . 55
4.2 a-Lecture hall partitions . . . . . . . . . . . . . . . . 59
4.3 Anti-lecture hall compositions . . . . . . . . . . . . . 60
4.4 Truncated objects . . . . . . . . . . . . . . . . . . . 89
4.5 (k; l)-Lecture hall partitions . . . . . . . . . . . . . . 91
Bibliography 93
Index 98
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