Combinatorics is a subject of increasing importance, owing to its links with computer science, statistics and algebra. This is a textbook aimed at second-year undergraduates to beginning graduates. It stresses common techniques (such as generating functions and recursive construction) which underlie the great variety of subject matter and also stresses the fact that a constructive or algorithmic proof is more valuable than an existence proof. The book is divided into two parts, the second at a higher level and with a wider range than the first. Historical notes are included which give a wider perspective on the subject. More advanced topics are given as projects and there are a number of exercises, some with solutions given.
Preface; 1. What is combinatorics?; 2. On numbers and counting; 3. Subsets, partitions, permutations; 4. Recurrence relations and generating functions; 5. The principle of inclusion and exclusion; 6. Latin squares and SDRs; 7. Extremal set theory; 8. Steiner triple theory; 9. Finite geometry; 10. Ramsey's theorem; 11. Graphs; 12. Posets, lattices and matroids; 13. More on partitions and permutations; 14. Automorphism groups and permutation groups; 15. Enumeration under group action; 16. Designs; 17. Error-correcting codes; 18. Graph colourings; 19. The infinite; 20. Where to from here?; Answers to selected exercises; Bibliography; Index.
