关于算法分析的这多卷论著已经长期被公认为经典计算机科学的定义性描述。这一册以及刚刚出版的第4卷第2册揭开了人们急切等待的《计算机程序设计艺术第4卷组合算法》的序幕。作为关于组合查找的冗长一章的一部分，这一册开始关于生成所有组合和分划的讨论。在Knuth讨论这两个主题的过程中，读者不仅会看到很多新内容，并且会发现本册与卷1至卷3及计算机科学和数学的其他方面的丰富联系。一如既往，书中包括了大量的习题和富有挑战性的难题。.
本册以及第4卷第2册的出版揭开了人们急切等待的《计算机程序设计艺术第4卷组合算法》的序幕。作为关于组合查找的冗长一章的一部分，这一册开始关于生成所有组合和分划的讨论。在Knuth讨论这两个主题的过程中，读者不仅会看到很多新内容，并且会发现本册与前三卷及计算机科学和数学的其他方面的丰富联系。一如既往，书中包括了大量的习题和富有挑战性的难题。通过讨论有关的游戏和数学难题，Knuth阐明一个重要的观点：严肃的程序设计也可以是一种乐趣。...

Preface.
Chapter 7 Combinatorial Searching
7.2 Generating All Possibilities
7.2.1 Generating Basic Combinatorial Patterns
7.2.1.1 Generating all n-tuples
7.2.1.2 Generating all permutations
7.2.1.3 Generating all combinations
7.2.1.4 Generating all partitions
7.2.1.5 Generating all set partitions
Answers to Exercises
译者序
前言..
第7章 组合查找
7．2 生成所有可能性
7．2．1 生成基本的组合模式
7．2．1．1 生成所有n元组
7．2．1．2 生成所有排列
7．2．1．3 生成所有组合
7．2．1．4 生成所有分划
7．2．1．5 生成所有集合的分划

THIS BOOKLET is Fascicle 3 of The Art of Computer Programming, Volume 4: Combinatorial Algorithms. As explained in the preface to Fascicle I of Volume 1, I'm circulating the material in this preliminary form because I know that the task of completing Volume 4 will take many years; I can't wait for people to begin reading what I've written so far and to provide valuable feedback. .
To put the material in context, this fascicle contains Sections 7.2.1.3, 7.2.1.4, and 7.2.1.5 of a long, long chapter on combinatorial searching. Chapter 7 will eventually fill three volumes (namely Volumes 4A, 4B, and 4C), assuming that I'm able to remain healthy. It will begin with a short review of graph theory, with emphasis on some highlights of significant graphs in The Stanford GraphBase, from which I will be drawing many examples. Then comes Section 7.1, which deals with bitwise manipulation and with algorithms relating to Boolean functions. Section 7.2 is about generating all possibilities, and it begins with Section 7.2.1: Generating Basic Combinatorial Patterns. Details about various useful ways to generate n-tuples appear in Section 7.2.1.1, and the generation of permutations is discussed in Section 7.2.1.2. That sets the stage for the main contents of the present booklet, namely Section 7.2.1.3 (which extends the ideas to combinations of n things taken t at a time); Section 7.2.1.4 (about partitions of an integer); and Section 7.2.1.5 (about partitions of a set). Then will come Section 7.2.1.6 (about trees) and Section 7.2.1.7 (about the history of combinatorial generation), in Fascicle 4. Section 7.2.2 will deal with backtracking in general. And so it will go on, if all goes well; an outline of the entire Chapter 7 as currently envisaged appears on the taocp webpage that is cited on page ii.
I had great pleasure writing this material, akin to the thrill of excitement that I felt when writing Volume 2 many years ago. As in Volume 2, where I found to my delight that the basic principles of elementary probability theory and number theory arose naturally in the study of algorithms for random number generation and arithmetic, I learned while preparing Section 7.2.1 that the basic principles of elementary combinatorics arise naturally and in a highly motivated way when we study algorithms for combinatorial generation. Thus, I found once again that a beautiful story was "out there" waiting to be told. ..
For example, in the present booklet we find many of the beautiful patterns formed by combinations, with and without repetition, and how they relate to famous theorems of extremal combinatorics. Then comes my chance to tell the extraordinary story of partitions; indeed, the theory of partitions is one of the nicest chapters in all of mathematics. And in Section 7.2.1.5, a littleknown triangle of numbers, discovered by C. S. Peirce, turns out to simplify and unify the study of set partitions, another vital topic. Along the way I've included expositions of two mathematical techniques of great importance in the analysis of algorithms: Poisson's summation formula, and the powerful saddle point method. There are games and puzzles too, as in the previous fascicles.
My original intention was to devote far less space to these subjects. But when I saw how fundamental the ideas were for combinatorial studies in general, I knew that I could never be happy unless I covered the basics quite thoroughly. Therefore I've done my best to build a solid foundation of theoretical and practical ideas that will support many kinds of reliable superstructures.
I thank Frank Ruskey for bravely foisting an early draft of this material on college students and for telling me about his classroom experiences. Many other readers have also helped me to check the first drafts; I wish to thank especially George Clements and Svante Janson for their penetrating comments.
I shall happily pay a finder's fee of $2.56 for each error in this fascicle when it is first reported to me, whether that error be typographical, technical, or historical. The same reward holds for items that I forgot to put in the index. And valuable suggestions for improvements to the text are worth 32c each. (Furthermore, if you find a better solution to an exercise, I'll actually reward you with immortal glory instead of mere money, by publishing your name in the eventual book:-)
Notations that are used here and not otherwise explained can be found in the Index to Notations at the end of Volumes 1, 2, or 3. Those indexes point to the places where further information is available. Of course Volume 4 will some day contain its own Index to Notations.
Machine-language examples in all future editions of The Art of Computer Programming will be based on the MMIX computer, which is described in Volume 1, Fascicle 1.
Cross references to yet-unwritten material sometimes appear as '00' in the following pages; this impossible value is a placeholder for the actual numbers to be supplied later. ...
Happy reading!
Stanford, California D. E. K.
June 2005