What Is Combinatorics Anyway?
Broadly speaking, combinatorics is the branch of mathematics dealing
with different ways of selecting objects from a set or arranging objects. It
tries to answer two major kinds of questions, namely, counting questions: how many ways can a selection or arrangement be chosen with a particular set of properties; and structural
questions: does there exist a selection or arrangement of objects with a
particular set of properties?
The authors have presented a text for students at all levels of preparation.
For some, this will be the first course where the students see several real proofs.
Others will have a good background in linear algebra, will have completed the calculus
stream, and will have started abstract algebra.
The text starts by briefly discussing several examples of typical combinatorial problems
to give the reader a better idea of what the subject covers. The next
chapters explore enumerative ideas and also probability. It then moves on to
enumerative functions and the relations between them, and generating functions and recurrences.,
Important families of functions, or numbers and then theorems are presented.
Brief introductions to computer algebra and group theory come next. Structures of particular
interest in combinatorics: posets, graphs, codes, Latin squares, and experimental designs follow. The
authors conclude with further discussion of the interaction between linear algebra
and combinatorics.
Features
- Two new chapters on probability and posets.
- Numerous new illustrations, exercises, and problems.
- More examples on current technology use
- A thorough focus on accuracy
- Three appendices: sets, induction and proof techniques, vectors and matrices, and biographies with historical notes,
- Flexible use of MapleTM and MathematicaTM
In Introduction to Combinatorics, Wallis (emer., Southern Illinois Univ.) and George (Gordon State College) present a well-thought-out compilation of topics covering elementary combinatorics. At the beginning, the authors present a thorough background on the fundamentals of combinatorics with topics such as permutations and combinations, the pigeonhole principle, and the principle of inclusion and exclusion. Later chapters are independent of one another and can be selected based on student and instructor interests. These topics include graph theory, coding theory, and balanced incomplete block designs. At the end of each chapter, there are exercises and problems. These vary in difficulty from straightforward practice to more involved proof problems. Solutions and/or hints are provided in the back of the book. In addition, three appendixes discuss proof techniques, matrices and vectors, and historical figures; these allow flexibility in covering the material in various ways that can be based on students' backgrounds. Overall, this textbook is a highly readable work that will benefit and enlighten all those interested in learning about combinatorics. It will work in a traditional classroom setting and for independent study. Given the level of material, it is geared toward junior or senior level undergraduate students.
--S. L. Sullivan, Catawba College
