Mathematics
Combinatorics
Combinatorics is a branch of mathematics that deals with counting and arranging objects. It involves the study of permutations, combinations, and other related concepts. Combinatorics has applications in various fields such as computer science, statistics, and physics.
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10 Key excerpts on "Combinatorics"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 7 Combinatorics Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of Combinatorics include counting the structures of a given kind and size ( enumerative Combinatorics ), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory ), finding largest, smallest, or optimal objects ( extremal Combinatorics and combinatorial optimization ), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems ( algebraic Combinatorics ). Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and Combinatorics also has many appli-cations in optimization, computer science, ergodic theory and statistical physics. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century however powerful and general theoretical methods were developed, making combi-natorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of Combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies Combinatorics is often referred to as a combinatorialist or (less frequently) combinatorist . ________________________ WORLD TECHNOLOGIES ________________________ History of Combinatorics An example of change ringing (with six bells) Basic combinatorial concepts and enumerative results have appeared throughout the ancient world.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 1 Combinatorics Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of Combinatorics include counting the structures of a given kind and size ( enumerative Combinatorics ), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory ), finding largest, smallest, or optimal objects ( extremal com-binatorics and combinatorial optimization ), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems ( algebraic Combinatorics ). Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and Combinatorics also has many appli-cations in optimization, computer science, ergodic theory and statistical physics. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century however powerful and general theoretical methods were developed, making combi-natorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of Combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies Combinatorics is often referred to as a combinatorialist or (less frequently) combinatorist . ________________________ WORLD TECHNOLOGIES ________________________ History of Combinatorics An example of change ringing (with six bells) Basic combinatorial concepts and enumerative results have appeared throughout the ancient world.
eBook - PDF- Paul Zeitz(Author)
- 2016(Publication Date)
- Wiley(Publisher)
6 Combinatorics 6.1 Introduction to Counting Combinatorics is the study of counting. That sounds rather babyish, but in fact counting problems can be quite deep and interesting and have many connections to other branches of mathematics. For example, consider the following problem. Example 6.1.1 (Czech and Slovak 1995) Decide whether there exist 10,000 10-digit numbers divisible by 7, all of which can be obtained from one another by a reordering of their digits. On the surface, it looks like a number theory problem. But it is actually just a question of carefully counting the correct things. We will solve this problem soon, on page 204, but first we need to develop some basic skills. Our first goal is a good understanding of the ideas leading up to binomial theorem. We assume that you have studied this subject a little bit before, but intend to review it and expand upon it now. Many of the concepts will be presented as a sequence of statements for you to verify before moving on. Please do not rush; make sure that you really understand each statement! In particular, pay attention to the tiniest of arithmetical details: Good combinatorial reasoning is largely a matter of knowing exactly when to add, multiply, subtract, or divide. Permutations and Combinations Items 6.1.2–6.1.12 introduce the concepts of permutations and combinations, and use only addition, multiplication, and division. 6.1.2 Simple Addition. If there are varieties of soup and varieties of salad, then there are + possible ways to order a meal of soup or salad (but not both soup and salad). 6.1.3 Simple Multiplication. If there are varieties of soup and varieties of salad, then there are possible ways to order a meal of soup and salad. 6.1.4 Let and be finite sets that are disjoint ( ∩ = ∅). Then 6.1.2 is equivalent to the statement | ∪ | = || + ||.
eBook - PDFDiscrete Mathematics
Mathematical Reasoning and Proof with Puzzles, Patterns, and Games
- Douglas E. Ensley, J. Winston Crawley(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
5 Combinatorics This chapter introduces the basic ideas of Combinatorics, one of the major ar- eas of discrete mathematics. Many people refer to Combinatorics questions as “counting problems.” Although this is an oversimplification, it is true enough of the topics that we will study in this chapter. We have of course seen counting problems in earlier chapters as we studied the sizes of various sets to better understand the nature of operations on sets and as we discussed the con- nection between properties of a function (like one-to-one and onto) and the relative sizes of the sets that comprised the function’s domain and codomain. Hence, Combinatorics is very much part of the other topics we have studied up to this point, and we will soon see that it is essential to the study of probability in the following chapter as well. We begin the present chapter with some traditional techniques for an- swering questions of the form “How many. . . ?” Along the way, we will en- counter some less traditional techniques such as recursive modeling applied to more difficult problems. These techniques can, in turn, be applied to other topics such as methods for finding closed formulas for recurrence relations. We end the chapter with methods for finding closed formulas for recursively defined sequences of the type that we encountered in the first two chapters of the book. In all this, an important ingredient in the study of Combinatorics is the representation of new objects or situations in terms of simpler objects. A large part of this is the ability to recognize when two problems are actually the same. This will become one of the most valuable skills you can take away from your study of Combinatorics. 368 5.1 Introduction 369 5.1 Introduction If you ever browse through a standard pocket dictionary, you will notice that the word “Combinatorics” does not appear within its pages. You may conclude that this is a word mathematicians made up on a slow day at the office.
eBook - PDFA Discrete Transition to Advanced Mathematics
Second Edition
- Bettina Richmond, Thomas Richmond(Authors)
- 2023(Publication Date)
- American Mathematical Society(Publisher)
Chapter 4 Combinatorics Whenever you can, count. —Sir Francis Galton (1822–1911) There are three kinds of mathematicians. Those who can count and those who can’t. —Anonymous 4.1. Getting from Point A to Point B Combinatorics is the branch of mathematics which deals with counting. By counting, we typically do not mean counting the number of objects in a tangible set, but counting 135 136 4. Combinatorics the number of possible outcomes in some mathematical setting. For instance, combi- natorics would more likely be used to count the number of distinct possibilities for the outcome of a five-card hand dealt from a standard deck than to count the number of cards in the deck. Our first section addresses the following counting problem. Problem 4.1.1. In the map shown in Figure 4.1.1, what is the length of the shortest route from point to point ? How many routes of this length are there from point to point ? ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ r A B r Figure 4.1.1. How many paths are there from A to B? For ease of reference, let us impose a coordinate system on the map by labeling the southeast thoroughfares 0th Avenue, 1st Avenue, 2nd Avenue, etc., and the southwest thoroughfares 0th Street, 1st Street, 2nd Street, etc. We will refer to the corner at - ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ sA B s 0 ℎ Avenue 1 Avenue 2 Avenue 3 Avenue 4 ℎ Avenue 0 ℎ Street 1 Street 2 Street 3 Street 4 ℎ Street ✛ ✛ ✛ ✛ 1 block from A 2 blocks from A 3 blocks from A 4 blocks from A ✠ ✠ ✠ ❅ ❅❘ ❅ ❅ ❘ ❅ ❅❘ ❅ ❅❘ ❅ ❅ ❘ ❅ ❅❘ ✠ ❅ ❅❘ ❅ ❅❘ ✠ ✠ Figure 4.1.2. Two paths from to . th Avenue and -th Street as the point (, ).
eBook - PDF- Thomas Koshy(Author)
- 2004(Publication Date)
- Academic Press(Publisher)
Chapter 6 Combinatorics and Discrete Probability The theory of probability affords an excellent illustration of the application of the theory of permutations and combinations which is the fundamental part of the algebra of discrete quantity. — G. CRYSTAL C ombinatorics is a fascinating branch of discrete mathematics, which deals with the art of counting. Very often we ask the question, In how many ways can a certain task be done ? Usually Combinatorics comes to our rescue. In most cases, listing the possibilities and counting them is the least desirable way of finding the answer to such a problem. Often we are not interested in enumerating the possibilities, but rather would like to know the total number of ways the task can be done. For instance, consider the following combinatorial problem: One type of automobile license plate number in Massachusetts consists of one letter and five digits. Find the number of such license plate numbers possible. • 5HO515 • • MASSACHUSETTS • Suppose you are willing to list all the possibilities and count them to find the answer. Assuming you know how to enumerate them systematically and that it would take a second to count each, it would take about 6 months to complete the counting alone. Obviously, this is an inefficient way to find the answer, especially when Combinatorics can do the job in seconds. (See Example 6.6.) A few other interesting problems we examine in this chapter are: • A secretary types up 10 different letters and 10 envelopes. In how many different ways can she place each letter in an envelope so that no letter is placed in the correct envelope? 343 344 Chapter 6 Combinatorics and Discrete Probability • Eleven guests would like to order soft drinks with their dinner. There are five choices for a soft drink: Coke Classic, Diet Coke, root beer, Pepsi, and Sprite. Find the number of different beverage selections possible.
eBook - PDFUse of Mathematical Literature
Information Sources for Research and Development
- A.R. Dorling(Author)
- 2014(Publication Date)
- Butterworth-Heinemann(Publisher)
7 Combinatorics N .L. Biggs andR.P. Jones* 7.1 INTRODUCTION It is only in recent years that combinatorial mathematics has attained some cohesion. Among the first practitioners of the subject were famous mathematicians such as Euler (who studied 'partitions') and Cayley (who studied 'trees'); other early workers were well-known mathematical eccentrics, such as Kirkman and Sylvester. The efforts of such men resulted in a subject full of fascinating oddities, but with no order or classification. A good idea of the state of the art at the end of the nineteenth century may be gained from three classic works on mathematical recreations: Lucas 1 , Rouse Ball 2 , and Ahrens 3 . Early attempts at organisation resulted in the books of Netto 4 and MacMahon 5 ; these books are still worth reading, but they give a very one-sided view of the subject as we know it today. In this survey the aim will be to introduce the reader to the main areas of combinatorial mathematics, with special emphasis on those which are centres of current research activity. There are several journals devoted specifically to this area of mathematics. The Journal of Combinatorial Theory (Academic Press, New York, 1966- ) has been published in two separate series since 1971: Series A contains papers on constructions, designs and * The authors are indebted to the following people for their helpful comments on a preliminary draft of this chapter: P.J. Cameron, A.W. Ingleton, J.H. van Lint, E.K. Lloyd, R. Rado, R.J. Wüson and D.R. Woodall. 96 Combinatorics 97 applications, while Series Β is devoted mainly to graph theory. Discrete Mathematics (North-Holland, Amsterdam, 1971- ) contains papers on all topics covered in this survey, and some associated subjects. In addition there are two recently founded Journals: Journal of Graph Theory (Wiley International, New York, 1977- ) anders Combina-toria (University of Waterloo, Waterloo, Ontario, 1975- ).
eBook - PDFQuantum Fields and Processes
A Combinatorial Approach
- John Gough, Joachim Kupsch(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
1 Introduction to Combinatorics How we count things turns out to have a powerful significance in physical problems! One of the oldest problems stems from undercounting and over- counting the number of possible configurations a particular system can have – mathematically, this is usually due to the fact that objects are mistakenly assumed to be indistinguishable when they are not, and vice versa. However, one of the great surprises of physics is that identical particles are funda- mentally indistinguishable. In this chapter, we will introduce some of the basic mathematical objects that occur in physical problems, and give their enumeration. Statistical mechanics is one of the key sources of ideas, so we spend some time on the basic concepts here, especially as partition functions are clear examples of generating functions that we will encounter later on. We will recall some of the basic mathematical concepts in enumeration, leading on to the role of generating functions. At the end, we make extensive use of generating functions, exploiting the methods for dealing with partition functions in statistical mechanics, but for specific combinatorial families such as permutations and partitions. We start, however, with the touchstone for all combinatorial problems: how to distribute balls in urns. 1.1 Counting: Balls and Urns Proposition 1.1.1 There are K N different ways to distribute the N distinguish- able balls among K distinguishable urns. The proof is based on the simple observation that there are K choices of urn for each of the N balls. Suppose next that we have more urns than balls. 1 2 Introduction to Combinatorics Figure 1.1 Occupation numbers of distinguishable balls in distinguishable urns. Proposition 1.1.2 The total number of ways to distribute the N distinguishable balls among K distinguishable urns so that no urn ends up with more than one ball is (later we will call this a falling factorial) K N K (K − 1) · · · (K − N + 1) .
eBook - PDF- Sheldon M. Ross(Author)
- 2000(Publication Date)
- Cambridge University Press(Publisher)
2. Combinatorial Analysis 2.1 Introduction Many problems can be solved simply by counting the number of differ-ent ways that a certain event can occur. In this chapter we show how one can efficiently do the counting in a variety of situations. In Sec-tion 2.2 we present the basic principle of counting, which is easily de-rived and extremely useful. Permutations are considered in Section 2.3 and combinations in 2.4. In Section 2.5, we consider the number of dif-ferent solutions of certain integral linear equalities. A counting method based on inclusions and exclusions is presented in Section 2.6, and one based on deriving and solving recursion equations is presented in Sec-tion 2.7. 2.2 The Basic Principle of Counting The following principle of counting will be basic to our work. Loosely put, it states that if an experiment consists of two parts, the first of which can result in any of m possible outcomes and the second in any of n pos-sible outcomes, then there are a total of mn possible outcomes of the experiment. Basic Principle of Counting Consider an experiment that consists of two phases. If the first phase can result in any of m possible outcomes and if, for each outcome of the first phase, there are n possible outcomes of the second phase, then there are a total of mn possible outcomes of the experiment. Proof. The basic principle can be proved by enumerating all possible outcomes of the experiment as follows: The Basic Principle of Counting 35 ( 1 , 1 ), ( 1 , 2 ), ..., ( 1 , n) ( 2 , 1 ), ( 2 , 2 ), ..., ( 2 , n) . . . (m, 1 ), (m, 2 ), ..., (m, n), where we say that the outcome is (i, j) if the first phase of the experi-ment results in its i th possible outcome and the next phase then results in the j th of its possible outcomes. Thus, the set of possible outcomes can be represented in m rows, each row containing n outcomes, which proves the result. Example 2.2a A women’s group consists of twelve women, each of whom has three children.
eBook - PDF- Richard Aufmann, Joanne Lockwood, Richard Nation, Daniel K. Clegg(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
Solving this problem is critical to enabling computers to interpret human speech. MATH MATTERS A standard deck of playing cards Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 700 CHAPTER 12 | Combinatorics and Probability Applying Several Counting Techniques The permutation formula is derived from the counting principle. This formula is a conve- nient way of expressing the number of ways in which the items in an ordered list can be arranged. Both the permutation formula and the counting principle are needed to solve some counting problems. EXAMPLE 4 Counting Using Several Methods Five women and four men are to be seated in a row of nine chairs. How many different seating arrangements are possible if a. there are no restrictions on the seating arrangements? b. the women sit together and the men sit together? Solution Because seating arrangements have a definite order, they are permutations. a. If there are no restrictions on the seating arrangements, then the number of seating arrangements is P( 9, 9). P( 9, 9) = 9! ( 9 − 9)! = 9! 0! = 9! = 362,880 There are 362,880 seating arrangements. b. This is a multi-stage experiment, so both the permutation formula and the counting principle will be used. There are 5! ways to arrange the women and 4! ways to arrange the men. We must also consider that either the women or the men could be seated at the beginning of the row. There are two ways to do this. By the counting principle, there are 2 ? 5! ? 4! ways to seat the women together and the men together.
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