Combinatorics of Compositions and Words Book

Introduction

Historical Overview—Compositions

Historical Overview—Words

A More Detailed Look

Basic Tools of the Trade

Sequences

Solving Recurrence Relations

Generating Functions

Compositions

Definitions and Basic Results (One Variable)

Restricted Compositions

Compositions with Restricted Parts

Connection between Compositions and Tilings

Colored Compositions and Other Variations

Research Directions and Open Problems

Statistics on Compositions

History and Connections

Subword Patterns of Length 2: Rises, Levels, and Drops

Longer Subword Patterns

Research Directions and Open Problems

Avoidance of Non-Subword Patterns in Compositions

History and Connections

Avoidance of Subsequence Patterns

Generalized Patterns and Compositions

Partially Ordered Patterns in Compositions

Research Directions and Open Problems

Words

History and Connections

Definitions and Basic Results

Subword Patterns

Subsequence Patterns—Classification

Subsequence Patterns—Generating Functions

Generalized Patterns of Type (2,1)

Avoidance of Partially Ordered Patterns

Research Directions and Open Problems

Automata and Generating Trees

History and Connections

Tools from Graph Theory

Automata

Generating Trees

The ECO Method

Research Directions and Open Problems

Asymptotics for Compositions

History

Tools from Probability Theory

Tools from Complex Analysis

Asymptotics for Compositions

Asymptotics for Carlitz Compositions

A Word on the Asymptotics for Words

Research Directions and Open Problems

Appendix A: Useful Identities and Generating Functions

Appendix B: Linear Algebra and Algebra Review

Appendix C: Chebychev Polynomials of the Second Kind

Appendix D: Probability Theory

Appendix E: Complex Analysis Review

Appendix F: Using Mathematica and Maple

Appendix G: C++ and Maple Programs

Appendix H: Notation

References

Exercises appear at the end of each chapter.