Combinatorics of Permutations. MiklosBona. Chapman & Hall/CRC, Boca Raton,Florida, 2004. ISBN 1-58488-434-7, Hardcover.400 pp., $89.95.
Permutations are a central topic in combi-natorics and have applications in many fields,such as sorting algorithms in computer scienceand permutation groups in group theory. Manymonographs study permutations, each with itsown emphasis and perspective. And many im-portant results in this area are spread acrossthe research literature of many different fields.This book aims to round up any topic relatedto the combinatorial nature of permutations andpresent it between one set of covers. For topicsthat are presented carefully in other texts, thecoverage is more of an overview, exposing thereader to the main ideas and then pointing theway where one can learn more. For topics thatare new, obscure, or neglected by more special-ized works, the author is more comprehensive.Throughout the book, there are frequent refer-ences to the excellent bibliography of more thantwo hundred research articles and books.
It is clear that the author finds his topic tobe full of serious fun. This enthusiasm is con-veyed in the conversational and engaging styleof the writing. The titles of the eight chaptersprovide an indication of this style, and of thetopics covered: In One Line and Close. Permu-tations as Linear Orders. Runs.; In One Lineand Anywhere. Permutations as Linear Orders.Inversions.; In Many Circles. Permutations asProducts of Cycles.; In Any Way But This. Pat-tern Avoidance. The Basics.; In This Way, ButNicely. Pattern Avoidance. Followup.; Meanand Insensitive. Random Permutations.; Permu-tations vs. Everything Else. Algebraic Combi-natorics of Permutations.; Get Them All. Al-gorithms and Permutations. Each chapter con-cludes with a set of about thirty or forty prob-lems of a theoretical nature (as opposed to beingcomputational), with solutions and notes for eachodd-numbered problem appearing in a section atthe end of the book. These problems are thenfollowed by a Problems Plus section contain-ing about ten or fifteen more difficult theoreticalproblems. These extra problems are followed im-mediately by a short discussion of each. Thesediscussions usually lead with a reference to a re-search article or monograph, which provides anindication of the difficulty level.
Unfortunately, it would appear that not muchcare was taken in proofreading this text. In the
limited amount of close reading performed by thisreviewer, there was an index reference that wastwo pages away from its target (logconcave),a figure that was inaccurate (Figure 7.3), a proofwhere a key explanation had the expression aireplaced by simply i (Theorem 2.3), a definitionwhose lead-in text directly contradicts the subse-quent definition (Definition 2.1), and a theoremwhose conclusion contains a typographical errorthat is obvious simply from the form of the state-ment (Theorem 1.4, on page 4!). Searching on theInternet did not locate any listings of errata.
This book is written to be used in a graduatelevel topics course. For that purpose it is ideallysuited. It would also be an excellent choice fora graduate student to use independently beforeinitiating a research program in this area, thoughsuch a student should be alerted to the possibil-ity of frequent errors. Experienced researchers incombinatorics will find the book useful as a guideto the literature on permutations. For graduatestudents with advanced interests in any field ofcombinatorics, the faculty who work with thesestudents, or the libraries that support them, thisbook is an excellent choice.
Robert A. BeezerUniversity of Puget Sound
An edited version of this review was published in
SIAM Review 47, No. 2 (2005) as part of the Book