Combinatorics of Permutations, Second Edition
By Miklos Bona
Chapman and Hall/CRC – 2012 – 478 pages
A Unified Account of Permutations in Modern Combinatorics
A 2006 CHOICE Outstanding Academic Title, the first edition of this bestseller was lauded for its detailed yet engaging treatment of permutations. Providing more than enough material for a one-semester course, Combinatorics of Permutations, Second Edition continues to clearly show the usefulness of this subject for both students and researchers and is recommended for undergraduate libraries by the MAA.
Much of the book has been significantly revised and extended. This edition includes a new section on alternating permutations and new material on multivariate applications of the exponential formula. It also discusses several important results in pattern avoidance as well as the concept of asymptotically normal distributions.
An entirely new chapter focuses on three sorting algorithms from molecular biology. This emerging area of combinatorics is known for its easily stated and extremely difficult problems, which sometimes can be solved using deep techniques from seemingly remote branches of mathematics.
Additional Exercises and Problems
All chapters in the second edition have more exercises and problems. Exercises are marked according to level of difficulty and many of the problems encompass results from the last eight years.
There is a new chapter nine, ‘devoted to sorting algorithms whose original motivation comes from molecular biology.’ Chapters 1, 3, 4, and 6 have been ‘significantly changed and expanded’ and all chapters have new exercises and problems, some of which reflect recent results. This is an excellent book on an important subject.
—Fernando Q. Gouvêa, MAA Reviews, December 2012 (This book is in the MAA's basic library list.)
How is the newcomer to this subject able to make sense of and sort out these bewildering possibilities? Until now it was necessary to consult a myriad of sources, from textbooks to journal articles, in order to grasp the whole picture. Now, however, Miklós Bóna has provided us with a comprehensive, engaging, and eminently readable introduction to all aspects of the combinatorics of permutations. The chapter on pattern avoidance is especially timely and gives the first systematic treatment of this fascinating and active area of research.
This book can be utilized at a variety of levels, from random samplings of the treasures therein to a comprehensive attempt to master all the material and solve all the exercises. In whatever direction the reader’s tastes lead, thorough enjoyment and appreciation of a beautiful area of combinatorics are certain to ensue.
—From the Foreword by Richard Stanley, MIT
Praise for the First Edition:
Winner of a 2006 CHOICE Outstanding Academic Title Award
One can easily imagine gems from this book forming the basis of a Martin Gardner-type column … the fascinating chapters here on pattern avoidance, particularly the formulation and proof of the Stanley-Wilf and Furedi-Hajnal conjectures, make this book essential … The author shows himself the master expositor, always efficient while never terse, ever the clairvoyant and generous anticipator of misreadings that might trip readers. Summing Up: Essential.
Throughout the book, there are frequent references to the excellent bibliography of more than two hundred research articles and books. It is clear that the author finds his topic to be full of ‘serious fun.’ This enthusiasm is conveyed in the conversational and engaging style of the writing … This book was written to be used in a graduate-level topics course. For that purpose it is ideally suited … Experienced researchers in combinatorics will find the book useful as a guide to the literature on permutations. For graduate students with advanced interests in any field of combinatorics, the faculty who work with these students, or the libraries that support them, this book is an excellent choice.
This advanced-level textbook grew out of a graduate combinatorics topics class. It serves that role quite well. … Applications are nicely sprinkled throughout the text. … a very extensive list of references with 208 entries … will be attractive to researchers in the area. This excellent text would serve a graduate seminar very well, but could also be used by advanced undergraduates who already have a background in combinatorics.
—Herbert E. Kasube, MAA Reviews
The literature on permutations is as extensive as permutations are manifold … What was missing until now was a comprehensive, up-to-date treatment of all aspects of the combinatorics of permutations … This is the first book which gives a systematic introduction to this fascinating and active area of research … All the subjects are presented in a very pleasant way: developments are always well motivated, explanations are transparent and illustrated by numerous examples. At the end of each chapter the reader finds a list of exercises, with detailed solutions … [containing] references [that] … are excellent starting points for further research.
—Zentralblatt für Mathematik
We found the author’s explanations very clear, and there is an abundance of useful examples and helpful figures . . . There is a rich bibliography for those seeking more information or full proofs of cited results.
—R. Gregory Taylor, SIGACT News, October 2008
[This book] was written by the author with love and enthusiasm for the subject and is a pleasure to read. Undergraduate and graduate students in combinatorics as well as researchers will find in it many interesting results and inspiring questions.
—Mathematical Reviews, 2005f
In One Line and Close. Permutations as Linear Orders.
In One Line and Anywhere. Permutations as Linear Orders. Inversions.
Inversion in Permutations of Multisets
In Many Circles. Permutations as Products of Cycles.
Decomposing a Permutation into Cycles
Type and Stirling Numbers
Cycle Decomposition versus Linear Order
Permutations with Restricted Cycle Structure
In Any Way but This. Pattern Avoidance. The Basics.
The Notion of Pattern Avoidance
Patterns of Length Three
Patterns of Length Four
The Proof of the Stanley–Wilf Conjecture
In This Way but Nicely. Pattern Avoidance. Follow-Up.
Containing a Pattern Many Times
Containing a Pattern a Given Number of Times
Mean and Insensitive. Random Permutations.
The Probabilistic Viewpoint
Variance and Standard Deviation
An Application: Longest Increasing Subsequences
Permutations versus Everything Else. Algebraic Combinatorics of Permutations.
The Robinson–Schensted–Knuth Correspondence
Posets of Permutations
Simplicial Complexes of Permutations
Get Them All. Algorithms and Permutations.
Stack Sorting Permutations
Variations of Stack Sorting
How Did We Get Here? Permutations as Genome Rearrangements.
Block Transpositions Revisited
Solutions to Odd-Numbered Exercises
List of Frequently Used Notation
Exercises, Problems, and Problem Solutions appear at the end of each chapter.
Miklós Bóna is a professor of mathematics at the University of Florida, where he is a member of the Academy of Distinguished Teaching Scholars. Dr. Bóna is an editor-in-chief of the Electronic Journal of Combinatorics. He has authored over 50 research articles and three combinatorics textbooks and has guided the research efforts of numerous undergraduate and graduate students in combinatorics. He earned a Ph.D. in mathematics from MIT.