Combinatorics
A Guided Tour
Published on 18. March 2010
Book
Hardback
410 pages
978-0-88385-762-5 (ISBN)
Description
Combinatorics is mathematics of enumeration, existence, construction, and optimization questions concerning finite sets. This text focuses on the first three types of questions and covers basic counting and existence principles, distributions, generating functions, recurrence relations, Polya theory, combinatorial designs, error correcting codes, partially ordered sets, and selected applications to graph theory including the enumeration of trees, the chromatic polynomial, and introductory Ramsey theory. The only prerequisites are single-variable calculus and familiarity with sets and basic proof techniques. 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. It also features approximately 350 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 470.
Reviews / Votes
This is a well-written, reader-friendly, and self-contained undergraduate course on combinatorics, focusing on enumeration. The book includes plenty of exercises, and about hal of them come with hints."" - M. Bona, Choice Magazine""The delineation of the topics is first rate-better than I have ever seen in any other book. Their presentation is generally thorough, with much of it in the form of worked problems. The book is very much designed as a textbook; there are plenty of problems at the end of each section. ... CAGT has both good breadth and great presentation; it is in fact a new standard in presentation for combinatorics, essential as a resource for any instructor, including those teaching out of a different text. For the student: If you are just starting to build a library in combinatorics, this should be your first book."" - The UMAP Journal
""Combinatorics is an excellent candidate for a special topics course for mathematics majors; with the broad spectrum of applications that course can simultaneously be an advanced and a capstone course. This book would be an excellent selection for the textbook of such a course. The explanations are at an appropriate level for the audience and there are exercises at the end of each section...The coverage is also sufficient of breadth; all of the major areas of combinatorics are covered...This book is the best candidate for a textbook in combinatorics that I have encountered."" - Charles Ashbacher
More details
Series
Language
English
Place of publication
Washington
United States
Target group
College/higher education
Professional and scholarly
Illustrations
7 Halftones, black and white; 44 Line drawings, black and white
Dimensions
Height: 261 mm
Width: 182 mm
Thickness: 26 mm
Weight
870 gr
ISBN-13
978-0-88385-762-5 (9780883857625)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Person
David R. Mazur is Associate Professor of Mathematics at Western New England College in Springfield, Massachusetts. He was born on October 23, 1971 in Washington, D.C. He received his undergraduate degree in Mathematics from the University of Delaware in 1993, and also won the Department of Mathematical Sciences' William D. Clark prize for 'unusual ability' in the major that year. He then received two fellowships for doctoral study in the Department of Mathematical Sciences (now the Department of Applied Mathematics and Statistics) at The Johns Hopkins University. From there he received his Master's in 1996 and his Ph.D. in 1999 under the direction of Leslie A. Hall, focusing on operations research, integer programming, and polyhedral combinatorics. His dissertation, 'Integer Programming Approaches to a Multi-Facility Location Problem', won first prize in the 1999 joint United Parcel Service/INFORMS Section on Location Analysis Dissertation Award Competition. The competition occurs once every two years to recognize outstanding dissertations in the field of location analysis. Professor Mazur began teaching at Western New England College in 1999 and received tenure and promotion to Associate Professor in 2005. He was a 2000-2001 Project NExT fellow and continues to serve this program as a consultant. He is an active member of the Mathematical Association of America, having co-organized several sessions at national meetings. He currently serves on the MAA's Membership Committee.
Content
Preface; Before you go; Notation; Part I. Principles of Combinatorics: 1. Typical counting questions, the product principle; 2. Counting, overcounting, the sum principle; 3. Functions and the bijection principle; 4. Relations and the equivalence principle; 5. Existence and the pigeonhole principle; Part II. Distributions and Combinatorial Proofs: 6. Counting functions; 7. Counting subsets and multisets; 8. Counting set partitions; 9. Counting integer partitions; Part III. Algebraic Tools: 10. Inclusion-exclusion; 11. Mathematical induction; 12. Using generating functions, part I; 13. Using generating functions, part II; 14. techniques for solving recurrence relations; 15. Solving linear recurrence relations; Part IV. Famous Number Families: 16. Binomial and multinomial coefficients; 17. Fibonacci and Lucas numbers; 18. Stirling numbers; 19. Integer partition numbers; Part V. Counting Under Equivalence: 20. Two examples; 21. Permutation groups; 22. Orbits and fixed point sets; 23. Using the CFB theorem; 24. Proving the CFB theorem; 25. The cycle index and Polya's theorem; Part VI. Combinatorics on Graphs: 26. Basic graph theory; 27. Counting trees; 28. Colouring and the chromatic polynomial; 29. Ramsey theory; Part VII. Designs and Codes: 30. Construction methods for designs; 31. The incidence matrix, symmetric designs; 32. Fisher's inequality, Steiner systems; 33. Perfect binary codes; 34. Codes from designs, designs from codes; Part VIII. Partially Ordered Sets: 35. Poset examples and vocabulary; 36. Isomorphism and Sperner's theorem; 37. Dilworth's theorem; 38. Dimension; 39. Moebius inversion, part I; 40. Moebius inversion, part II; Bibliography; Hints and answers to selected exercises.