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Combinatorics of Permutations


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Table of Contents

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 MultisetsIn 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 StructureIn 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 ConjectureIn This Way but Nicely. Pattern Avoidance. Follow-Up.
Polynomial Recurrences
Containing a Pattern Many Times
Containing a Pattern a Given Number of TimesMean and Insensitive. Random Permutations.
The Probabilistic Viewpoint
Variance and Standard Deviation
An Application: Longest Increasing SubsequencesPermutations versus Everything Else. Algebraic Combinatorics of Permutations.
The Robinson-Schensted-Knuth Correspondence
Posets of Permutations
Simplicial Complexes of PermutationsGet Them All. Algorithms and Permutations.
Generating Permutations
Stack Sorting Permutations
Variations of Stack SortingHow Did We Get Here? Permutations as Genome Rearrangements.
Block Transpositions
Block Interchanges
Block Transpositions RevisitedSolutions to Odd-Numbered Exercises References List of Frequently Used Notation Index Exercises, Problems, and Problem Solutions appear at the end of each chapter.

About the Author

Miklos Bona is a professor of mathematics at the University of Florida, where he is a member of the Academy of Distinguished Teaching Scholars. Dr. Bona 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.


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. --CHOICE 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 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 fur 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

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