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48202 results in Computer Science

Page 2332 of 2411

  • Preface

  • Peter J. Cameron
  • Book:
    Combinatorics
    Published online:
    28 May 2018
    Print publication:
    06 October 1994, pp ix-x

  • Summary

    Ive got to work the E qwations and the low cations Ive got to comb the nations of it.

    Russell Hoban, Riddley Walker (1980)

    We have not begun to understand the relationship between combinatorics and conceptual mathematics.

    J. Dieudonné, A Panorama of Pure Mathematics (1982)

    If anything at all can be deduced from the two quotations at the top of this page, perhaps it is this: Combinatorics is an essential part of the human spirit; but it is a difficult subject for the abstract, axiomatising Bourbaki school of mathematics to comprehend. Nevertheless, the advent of computers and electronic communications have made it a more important subject than ever.

    This is a textbook on combinatorics. It's based on my experience of more than twenty years of research and, more specifically, on teaching a course at Queen Mary and Westfield College, University of London, since 1986. The book presupposes some mathematical knowledge. The first part (Chapters 2–11) could be studied by a second-year British undergraduate; but I hope that more advanced students will find something interesting here too (especially in the Projects, which may be skipped without much loss by beginners). The second half (Chapters 12–20) is in a more condensed style, more suited to postgraduate students.

    I am grateful to many colleagues, friends and students for all kinds of contributions, some of which are acknowledged in the text; and to Neill Cameron, for the illustration on p. 128.

    I have not provided a table of dependencies between chapters. Everything is connected; but combinatorics is, by nature, broad rather than deep. The more important connections are indicated at the start of the chapters.

  • 1 - What is Combinatorics?

  • Peter J. Cameron
  • Book:
    Combinatorics
    Published online:
    28 May 2018
    Print publication:
    06 October 1994, pp 1-6

  • Summary

    Combinatorics is the slums of topology.

    J. H. C. Whitehead (attr.)

    I have to admit that he was not bad at combinatorial analysis — a branch, however, that even then I considered to be dried up.

    Stanislaw Lem, His Master's Voice (1968)

    Combinatorics is special. Most mathematical topics which can be covered in a lecture course build towards a single, well-defined goal, such as Cauchy's Theorem or the Prime Number Theorem. Even if such a clear goal doesn't exist, there is a sharp focus (finite groups, perhaps, or non-parametric statistics). By contrast, combinatorics appears to be a collection of unrelated puzzles chosen at random.

    Two factors contribute to this. First, combinatorics is broad rather than deep. Its tentacles stretch into virtually all corners of mathematics. Second, it is about techniques rather than results. As in a net, threads run through the entire construction, appearing unexpectedly far from where we last saw them. A treatment of combinatorics which neglects this is bound to give a superficial impression.

    This feature makes the teacher's job harder. Reading, or lecturing, is inherently one-dimensional. If we follow one thread, we miss the essential interconnectedness of the subject.

    I have attempted to meet this difficulty by various devices. Each chapter begins with a list of topics, techniques, and algorithms considered in the chapter, and cross-references to other chapters. Also, some of the material is set in smaller type and can be regarded as optional.

Page 2332 of 2411