Combinatorics: A Guided Tour

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The text emphasizes the brands of thinking that are characteristic of combinatorics: bijective and combinatorial proofs, recursive analysis, and counting problem classification. It is flexible enough to be used for undergraduate courses in combinatorics, second courses in discrete mathematics, introductory graduate courses in applied mathematics programs, as well as for independent study or reading courses. What makes this text a guided tour are the approximately reading questions spread throughout its eight chapters.

These questions provide checkpoints for learning and prepare the reader for the end-of-section exercises of which there are over Most sections conclude with Travel Notes that add color to the material of the section via anecdotes, open problems, suggestions for further reading, and biographical information about mathematicians involved in the discoveries.

Access codes may be used or not included. We use UPS Ground for all shipments. PO Boxes could delay shipment. Login to see store details. Ships Fast. Published In: United States, 18 March Combinatorics is mathematics of enumeration, existence, construction, and optimization questi Order now and we'll deliver when available. We'll e-mail you with an estimated delivery date as soon as we have more information. Your credit card will not be charged until we ship the item. S Edition otherwise as stated. Returns on textbooks: Returns only on books, absolutely no access codes with be returned for any reason.

Notes: Textbooks may not include supplemental items. The coverage of generating functions includes techniques for solving recurrence relations. In Chapter 4 we use the techniques of the previous chapters to give a more in-depth study of the binomial and multinomial coefficients, Fibonacci numbers, Stirling numbers of the first and second kinds, and integer partition numbers. In Chapter 5 we cover counting problems involving equivalence and symmetry consid- erations.

MATH BC Combinatorics

The main results are the Cauchy-Frobenius-Burnside theorem and Polyas enu- meration theorem. Though Polyas theorem arose from an application to the enumeration of chemical compounds, it has since proved to be a powerful and versatile tool in all sorts of other applications. We begin this chapter by introducing those aspects of group theory necessary to understand the theorems, and then give many illustrations of how to apply them. In Chapter 6 we give a short survey of some combinatorial problems in graph theory.

These include the enumeration of labeled trees and binary search trees, coloring and the chromatic polynomial, and introductory Ramsey theory. Though Ramsey theory can be introduced without the aid of graphs, the edge-coloring interpretation is convenient and concrete. The first section of this chapter covers basic graph theory concepts for the reader who is unfamiliar with graphs.

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In Chapter 7 we cover two of the most compelling applications of combinatorics: com- binatorial designs and error-correcting codes. As a bonus, the mathematical questions sur- rounding these applications are just as compelling if not more so. In the three sections on designs we cover existence and construction methods, symmetric designs, and triple systems.

In the two sections on error-correcting codes, we construct the family of binary Hamming codes and derive their error-correcting properties, study the interplay between codes and designs, and discuss the truly astonishing results concerning the existence of perfect codes. In Chapter 8 we conclude our journey by studying relations that are, in some sense, lurking behind much of combinatorics: partially ordered sets or posets.

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We study some classical results Sperners theorem and Dilworths theorem and also the concept of poset dimension. In the final two sections we introduce the theory of Mobius inversion and do so with a two-fold purpose: to provide a unifying framework for several combinatorial ideas and to prepare the reader for further study. There are several important topics not included on the tour. The coverage of graph theory in Chapter 6, though it contains an introductory section, is focused fairly narrowly on the topics mentioned earlier.

A major branch of combinatorics, namely combinatorial optimization, is left out entirely. Also, the coverage of designs and codes is driven by the particular applications. As such, we do not cover projective planes, combinatorial geome- tries, or Latin squares.

MAA Textbooks: Combinatorics : A Guided Tour 17 by David R. Mazur (2010, Hardcover)

Features of this book Reading questions. What makes this book a guided tour are the approximately Ques- tions spread throughout the eight chapters. These allow the reader to be an active partici- pant in the discussion and are meant to provide a more honest reflection of the process by which we all learn mathematics. Reading a math book without pencil and paper in hand is like staying in your hotel and viewing the interesting sites from your window.

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Youll get more out of the tour if you leave the hotel and go explore on foot. To count the cows in a field you could either 1 count their heads, or 2 count their legs and divide by 4.

In a combinatorial proof one asks a count- ing question and then answers it correctly using two different approaches. This little idea leads to some beautiful,memorable, and even fun! Wherever possible, we present combinatorial proofs because they promote understanding and build combinatorial think- ing skills. Classification of counting problems. The hard part about counting is determining the type of objects being counted.

Instead of covering lists, lists without repetition permu- tations , subsets combinations , and multisets combinations with repetition in separate sections, in Section 1. Conversational style, some big examples. Ive tried to maintain a conversational and somewhat informal tone throughout the book. This occasionally means that brevity is sac- rificed for the sake of clarity. In certain situations Ive included bigger examples when helpful. For two examples see Figure 3. Links with continuous mathematics. At appropriate places in the text Ive highlighted where calculus, differential equations, linear algebra, etc.

These help dispel the notion that combinatorics is a discrete-only field. Instructor flexibility. Completion of the reading questions prior to class frees the in- structor from lecturing on basic material. Class time could then be used to clarify difficul- ties, lecture on advanced topics, have a problem session, or assign group work.

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This also allows class time for reviewing proof techniques, linear algebra, power series, or modular arithmetic, if necessary. See below for optional prerequisites. Courses and ways to use this book This book has two primary uses. Guided Tour Chaperone Checklist -? Enumerative Combinatorics Through Guided Discovery?

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