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Combinatorics of Permutations, Second Edition

By Miklos Bona

Chapman and Hall/CRC – 2012 – 478 pages

Series: Discrete Mathematics and Its Applications

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    June 10th 2012


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.

Expanded Chapters

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.

New Chapter

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.


The extended new edition of Bóna’s textbook will continue to provide a wealth of information on the combinatorial facet of permutations, to both students and researchers in the subject.

— Martin Kiazar, in Mathematical Review Clippings

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.

SIAM Review

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, 2005


In One Line and Close. Permutations as Linear Orders.


Alternating Runs

Alternating Subsequences

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

Monotone Patterns

Patterns of Length Four

The Proof of the Stanley–Wilf Conjecture

In This Way but Nicely. Pattern Avoidance. Follow-Up.

Polynomial Recurrences

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.

Generating Permutations

Stack Sorting Permutations

Variations of Stack Sorting

How Did We Get Here? Permutations as Genome Rearrangements.


Block Transpositions

Block Interchanges

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.

Author Bio

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.

Name: Combinatorics of Permutations, Second Edition (Hardback)Chapman and Hall/CRC 
Description: By Miklos Bona. A Unified Account of Permutations in Modern CombinatoricsA 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...
Categories: Combinatorics, Intelligent Systems, Algorithms & Complexity