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Combinatorics

Combinatorics
A Guided Tour

$70.00 (P)

Part of Mathematical Association of America Textbooks

  • Date Published: March 2010
  • availability: This item is not supplied by Cambridge University Press in your region. Please contact Mathematical Association of America for availability.
  • format: Hardback
  • isbn: 9780883857625

$ 70.00 (P)
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About the Authors
  • 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, Pólya 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.

    • 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
    • 350 reading questions are spread through the chapters, providing checkpoints for learning to prepare the reader for the end-of-section exercises
    • Travel notes enrich the material of each section with anecdotes, open problems, suggestions for further reading and biographical information about mathematicians involved in the discoveries
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    Product details

    • Date Published: March 2010
    • format: Hardback
    • isbn: 9780883857625
    • length: 410 pages
    • dimensions: 261 x 182 x 26 mm
    • weight: 0.87kg
    • contains: 51 b/w illus.
    • availability: This item is not supplied by Cambridge University Press in your region. Please contact Mathematical Association of America for availability.
  • Table of Contents

    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 Pólya'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. Möbius inversion, part I
    40. Möbius inversion, part II
    Bibliography
    Hints and answers to selected exercises.

  • Resources for

    Combinatorics

    David R. Mazur

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  • Author

    David R. Mazur, Western New England College, Massachusetts
    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.

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