離散數(shù)學(xué)引論

前言

在學(xué)校教書多年，當(dāng)學(xué)生（特別是本科生）問(wèn)有什么好的參考書時(shí)，我們所能推薦的似乎除了教材還是教材，而且不同教材之間的差別并不明顯、特色也不鮮明。所以多年前我們就開始醞釀，希望為本科學(xué)生引進(jìn)一些好的參考書，為此清華大學(xué)數(shù)學(xué)科學(xué)系的許多教授與清華大學(xué)出版社共同付出了很多心血。這里首批推出的十余本圖書，是從Springer出版社的多個(gè)系列叢書中精心挑選出來(lái)的。在叢書的籌劃過(guò)程中，我們挑選圖書最重要的標(biāo)準(zhǔn)并不是完美，而是有特色并包容各個(gè)學(xué)派（有些書甚至有爭(zhēng)議，比如從數(shù)學(xué)上看也許不夠嚴(yán)格），其出發(fā)點(diǎn)是希望我們的學(xué)生能夠吸納百家之長(zhǎng)；同時(shí)，在價(jià)格方面，我們也做了很多工作，以使得本系列叢書的價(jià)格能讓更多學(xué)校和學(xué)生接受，使得更多學(xué)生能夠從中受益。本系列圖書按其定位，大體有如下四種類型（一本書可以屬于多類，但這里限于篇幅不能一一介紹）。

內(nèi)容概要

　　《離散數(shù)學(xué)引論》以簡(jiǎn)潔和通俗的形式介紹組合數(shù)學(xué)的一些本質(zhì)性內(nèi)容圖論的重要問(wèn)題，計(jì)數(shù)方法和試驗(yàn)設(shè)計(jì)，其中圖論約占一半篇幅?！峨x散數(shù)學(xué)引論》很適于和中國(guó)中學(xué)數(shù)學(xué)教材的內(nèi)容相銜接，閱讀《離散數(shù)學(xué)引論》所需的預(yù)備知識(shí)只是中學(xué)數(shù)學(xué)（唯一的例外是在圖論中需要矩陣的描述方式，但即使沒(méi)有學(xué)過(guò)線性代數(shù)，也是可以接受的）?！　杏写罅苛?xí)題和例題，習(xí)題附有部分解答和提示，適于自學(xué)?！峨x散數(shù)學(xué)引論》可用作數(shù)學(xué)、計(jì)算機(jī)科學(xué)、信息科學(xué)等專業(yè)大學(xué)本科生的組合數(shù)學(xué)教材，可在大學(xué)一年級(jí)講授。

作者簡(jiǎn)介

作者：(美國(guó))安德遜(Ian Anderson)

書籍目錄

Contents1. Counting and Binomial Coefficients1.1 Basic Principles1.2 Factorials1.3 Selections1.4 Binomial Coefficients and Pascal's Triangle 1.5 Selections with——Repetitions1.6 AUsefulMatrixInversion2. Recurrence2.1 Some Examples 2.2 The Auxiliary Equation Method2.3 Generating Fhnctions2.4 Derangements2.5 Sorting Algorithms2.6 Catalan Numbers3. Introduction to Graphs3.1 The Concept of a Graph3.2 Paths in Graphs3.3 Trees 3.4 Spanning Trees3.5 Bipartite Graphs3.t5 Planarity3.7 Polyhedra.4. Travelling Round a Graph4 1 Hamiltonian Graphs4.2 Planarity and Hamiltonian Graphs4.3 The Travelling Salesman Problem 4.4 Gray Codes4.5 EulerianDigraphs5. Partitions and Colourings5.1 Partitions of a Set 5.2 StirlingNumbers5.3 Counting Functions5.4 Vertex Colourings of Graphs 5.5 Edge Colourings of Graphs 6 The Inclusion-Exclusion Principle6.1 The Principle 6.2 Counting Surjections6.3 Counting Labelled Trees6.4 Scrabble.15.5 The MSnage Problem 7. Latin Squares and Hall's Theorem.7.1 Latin-Squares and -Orthogonality 7.2 Magic Squares7.3 Systems of Distinct Representatives7.4 From Latin Squares to Affine Planes8 Schedules and 1-Factorisations8.1 The Circle Method8.2 Bipartite Tournaments and 1-Factorisations of Kn8.3 Tournaments from Orthogonal Latin Squares9. Introduction to Designs. 9.1 Balanced Incomplete Block Designs9.2 Resolvable Designs 9.3 Finite Projective Planes9.4 Hadamard Matrices and Designs9.5 Difference Methods9.5 Hadamard Matrices and CodesAppendixSolutions Further ReadingBibliographyIndex

章節(jié)摘錄

插圖：The How, When, and Why of MathematicsWhat is mathematics？ Many people think of mathematics （incorrectly） as addition, subtraction, multiplication, and division of numbers. Those with more mathematical training may think of it as dealing with algorithms. But most professional mathematicians think of it as much more than that. While we certainly hope that our students will perform algorithms correctly, what we really want is for them to understand three things: how you do something, why it works, and when it works. The problems we present to you in this book concentrate on these three goals. If this is the first time you have been asked to prove theorems, you may find this to be quite a challenge. Not only will you be learning how to solve the problem, you will also be learning how to write up the solution. The necessary definitions and background to understand a problem, as well as a general plan of attack, will always be presented in the text. It's up to you to spend the time reading, trying various approaches, rereading, and reproaching. You will probably be spending more time on fewer exercises than you ever have before. While you are now beyond the stage of being given steps to follow and practice, there are general rules that can assist you in your transition to doing higher mathematics. Many people have written about this subject before.

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