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Combinatorics : Topics, Techniques, Algorithms - Peter J. Cameron

Combinatorics

Topics, Techniques, Algorithms

Paperback

Published: 5th December 1994
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Combinatorics is a subject of increasing importance because of its links with computer science, statistics, and algebra. This textbook stresses common techniques (such as generating functions and recursive construction) that underlie the great variety of subject matter, and the fact that a constructive or algorithmic proof is more valuable than an existence proof. The author emphasizes techniques as well as topics and includes many algorithms described in simple terms. The text should provide essential background for students in all parts of discrete mathematics.

"Cameron covers an impressive amount of material in a relatively small space...an outstanding supplement to other texts..." M. Henle, Choice "...used as a text at the senior or graduate level and is an excellent reference...The range of topics is very good." The UMAP Journal

Preface
What is Combinatorics?p. 1
Sample problems
How to use this book
What you need to know
Exercises
On numbers and countingp. 7
Natural numbers and arithmetic
Induction
Some useful functions
Orders of magnitude
Different ways of counting
Double counting
Appendix on set notation
Exercises
Subsets, partitions, permutationsp. 21
Subsets
Subsets of fixed size
The Binomial Theorem and Pascal's Triangle
Project: Congruences of binomial coefficients
Permutations
Estimates for factorials
Selections
Equivalence and order
Project: Finite topologies
Project: Cayley's Theorem on trees
Bell numbers
Generating combinatorial objects
Exercises
Recurrence relations and generating functionsp. 49
Fibonacci numbers
Aside on formal power series
Linear recurrence relations with constant coefficients
Derangements and involutions
Catalan and Bell numbers
Computing solutions to recurrence relations
Project: Finite fields and QUICKSORT
Exercises
The Principle of Inclusion and Exclusionp. 75
PIE
A generalisation
Stirling numbers
Project: Stirling numbers and exponentials
Even and odd permutations
Exercises
Latin squares and SDRsp. 87
Latin squares
Systems of distinct representatives
How many Latin squares?
Quasigroups
Project: Quasigroups and groups
Orthogonal Latin squares
Exercises
Extremal set theoryp. 99
Intersecting families
Sperner families
The De Bruijn-Erdos Theorem
Project: Regular families
Exercises
Steiner triple systemsp. 107
Steiner systems
A direct construction
A recursive construction
Packing and covering
Project: Some special Steiner triple systems
Project: Tournaments and Kirkman's schoolgirls
Exercises
Finite geometryp. 123
Linear algebra over finite fields
Gaussian coefficients
Projective geometry
Axioms for projective geometry
Projective planes
Other kinds of geometry
Project: Coordinates and configurations
Project: Proof of the Bruck-Ryser Theorem
Finite fields
Exercises
Ramsey's Theoremp. 147
The Pigeonhole Principle
Some special cases
Ramsey's Theorem
Bounds for Ramsey numbers
Applications
The infinite version
Exercises
Graphsp. 159
Definitions
Trees and forests
Minimal spanning trees
Eulerian graphs
Hamiltonian graphs
Project: Gray codes
The Travelling Salesman
Digraphs
Networks
Menger, Konig and Hall
Diameter and girth
Project: Moore graphs
Exercises
Posets, lattices and matroidsp. 187
Posets and lattices
Linear extensions of a poset
Distributive lattices
Aside on propositional logic
Chains and antichains
Products and dimension
The Mobius function of a poset
Matroids
Project: Arrow's Theorem
Exercises
More on partitions and permutationsp. 209
Partitions, diagrams and conjugacy classes
Euler's Pentagonal Numbers Theorem
Project: Jacobi's Identity
Tableaux
Symmetric polynomials
Exercises
Automorphism groups and permutation groupsp. 225
Three definitions of a group
Examples of groups
Orbits and transitivity
The Schreier-Sims algorithm
Primitivity and multiple transitivity
Examples
Project: Cayley digraphs and Frucht's Theorem
Exercises
Enumeration under group actionp. 245
The Orbit-counting Lemma
An application
Cycle index
Examples
Direct and wreath products
Stirling numbers revisited
Project: Cycle index and symmetric functions
Exercises
Designsp. 257
Definitions and examples
To repeat or not to repeat
Fisher's Inequality
Designs from finite geometry
Small designs
Project: Hadamard matrices
Exercises
Error-correcting codesp. 271
Finding out a liar
Definitions
Probabilistic considerations
Some bounds
Linear codes; Hamming codes
Perfect codes
Linear codes and projective spaces
Exercises
Graph colouringsp. 291
More on bipartite graphs
Vertex colourings
Project: Brooks' Theorem
Perfect graphs
Edge colourings
Topological graph theory
Project: The Five-colour Theorem
Exercises
The infinitep. 307
Counting infinite sets
Konig's Infinity Lemma
Posets and Zorn's Lemma
Ramsey theory
Systems of distinct representatives
Free constructions
The random graph
Exercises
Where to from here?p. 325
Computational complexity
Some graph-theoretic topics
Computer software
Unsolved problems
Further reading
Answers to selected exercisesp. 339
Bibliographyp. 343
Indexp. 347
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521457613
ISBN-10: 0521457610
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 368
Published: 5th December 1994
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 24.7 x 17.4  x 2.5
Weight (kg): 0.61