WINNER of a CHOICE Outstanding Academic Title Award for 2006! As linear orders, as elements of the symmetric group, modeled by matrices, modeled by graphs!permutations are omnipresent in modern combinatorics. They are omnipresent but also multifaceted, and while several excellent books explore particular aspects of the subject, no one book has covered them all. Even the classic results are scattered in various resources. Combinatorics of Permutations offers the first comprehensive, up to date treatment of both enumerative and extremal combinatorics and looks at permutation as linear orders and as elements of the symmetric group. The author devotes two full chapters to the young but active area of pattern avoidance. He explores the quest for the Stanley-Wilf conjecture and includes the recent and spectacular Marcus-Tardos proof of this problem. He examines random permutations and Standard Young Tableaux and provides an overview of the very rich algebraic combinatorics of permutations. The final chapter takes an in-depth look at combinatorial sorting algorithms. The author's style is relaxed, entertaining, and clearly reflects his enthusiasm for the "serious fun" the subject holds. Filled with applications from a variety of fields and exercises that draw upon recent research results, this book serves equally well as a graduate-level text and a reference for combinatorics researchers.

Table of Contents:
IN ONE LINE AND CLOSE. PERMUTATIONS AS LINEAR ORDERS. RUNS Descents Alternating Runs IN ONE LINE AND ANYWHERE. PERMUTATIONS AS LINEAR ORDERS. INVERSIONS 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; THE FOLLOW-UP Polynomial Recursions Containing a Pattern Many Times Containing a Pattern a Given Number of Times MEAN AND INSENSITIVE: RANDOM PERMUTATIONS The Probabilistic Viewpoint Expectation Variance and Standard Deviation An Application: Longest Increasing Subsequences PERMUTATIONS vs. 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 SOLUTIONS TO ODD-NUMBERED EXERCISES REFERENCES LIST OF FREQUENTLY USED NOTATIONS INDEX Each chapter also includes Exercises, Problems Plus, and Solutions to Problems Plus sections