Review of Bijective Combinatorics by Nicholas Loehr

Review by Miklós Bóna When I first heard that there is a book entitled " Bijective Combinatorics " , I was surprised. As all combinatorial enumerators, I love bijective proofs. But an entire book on them, with 570 pages? What kind of book would that be? A collection of the nicest known bijective proofs could not be quite that long. The short answer is that the title is somewhat misleading. This is an informative textbook on Enumerative Combinatorics, in which bijective proofs certainly get their fair share. Another central question that a reviewer must answer about the book he reviews is " just what kind of book is it " , in the sense of level, targeted audience, and style of presentation. That question is particularly difficult to answer for this book, since it can be viewed as several books in one. In the first four chapters, you find all the topics that you would find in other introductory textbooks on combinatorial enumeration, such as counting permutations, words, subsets, with or without repetitions, graphical enumeration, the formula of inclusion-exclusion, and the Möbius inversion formula. The discussion is somewhat deeper than what one finds in undergraduate textbooks , though probably not quite as advanced as that of the classic graduate books. There are plenty of exercises. Chapter 5, Ranking and Unranking, is more interesting. This reviewer has not seen this topic in any other combinatorics textbooks, and suspects that theoretical computer scientists will be more familiar with the subject than most combinatorialists. If certain combinatorial objects, such as permutations of length 25, are listed in some order, and we are asked where in that lists a given permutation p is located, then we are in fact asked what the rank of that permutation is in the ranking specified by this listing. It goes without saying that we would like to find a method that involves as few steps as possible, so going through the whole list is not a desirable option. This family of problems is called ranking. Not surprisingly, unranking is the procedure of computing the ith element of a list from the rules that were used to create that list. After defining these intriguing goals, the chapter continues with a rather long list of introductory results, then finally reaches some interesting instances of the problem, such as ranking in the set of n n−2 trees on n labeled vertices, …