Undergraduate Texts in Mathematics. New York etc.: Springer-Verlag. X, 183 p. DM 48.00 (1986).
This volume is the amplification of course notes in the area of ’constructive combinatorics’. This area includes the enumeration and listing of various kinds of finite structures as well as use of constructive arguments to prove theorems.
The text is suitable for an undergraduate level, involving few if any prerequisites. Further, it covers an attractive selection of topics and is full of stimulating questions and clever arguments that will be inspirational to many mathematics students.
A strong point is the set of exercises which are numerous and at various levels of challenge. There is also an appendix containing programs which implement some of the algorithms discussed in the text.
The subject matter is divided into four chapters; the first concerns enumeration of basic structures, such as permutations and partitions of sets and of integers, with emphasis on actual listing of such entities. The second concerns partially ordered sets including application to the Littlewood-Offord problem. A section on bijections, including discussion of the Prüfer Correspondence and Schensted’s Correspondence and tableaux is the largest chapter. Finally, a chapter entitled ’Involutions’ contains discussions of such topics as Vandermonde’s determinant, the Cayley-Hamilton theorem and the matrix-tree theorem, and of lattice paths.
This subject matter has the virtue that one can get into non-trivial mathematical results without bogging down in definitions or requiring lots of prior knowledge. It is an ideal ground therefore, for a student to be exposed to in order to realize that she or he loves mathematics and can relate to it. I therefore think that courses in this area should be encouraged. This is an excellent text for such a course.