This monograph requires no prior knowledge of combinatorial mathematics. In Chapter 1 we deal with the elementary properties of sets and define permutation, combination, and binomial coefficient. Of course we treat these concepts from a mature point of view, and from the outset we assume an appreciation for the subtleties of mathematical reasoning. Combinatorial mathematics is best studied within the framework of modern algebra, and for this reason we presuppose a certain familiarity with a few algebraic concepts. Matrices are the really important tool. They occur throughout the monograph and unify the various chapters. At first they are primarily rectangular arrays and little is needed in the way of background. Later they play a fuller role, and we apply the standard rules of matric manipulation. Number theory is used sparingly. An understanding of integral congruences is adequate for most purposes. Groups and fields are mentioned in passing. Only on rare occasions do we call for something beyond the definitions of these systems.
Many of our proofs rely on counting arguments, finite induction, or some other time-tested device. But this does not mean that combinatorial mathematics is easy. The subject is demanding and its exposition is troublesome. Our definitions and proofs are concise and they deserve careful scrutiny. But effort and ingenuity lead to mastery, and our subject holds rich rewards for those who learn its secrets.
We pursue certain topics with thoroughness and reach the frontiers of present-day research. But we pay a price for this and must omit much that is of interest. Each chapter contains its separate bibliography. These are guides for further study and do not aim at completeness. We also discuss in the pages that follow some vital questions that remain unanswered. Combinatorial mathematics is tremendously alive at this moment, and we believe that its greatest truths are still to be revealed.