Heubach / Mansour Combinatorics of Compositions and Words

Erscheinungsjahr 2010
ISBN: 978-1-4200-7268-6
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 504 Seiten

Reihe: Discrete Mathematics and Its Applications

A One-Stop Source of Known Results, a Bibliography of Papers on the Subject, and Novel Research Directions
Focusing on a very active area of research in the last decade, Combinatorics of Compositions and Words provides an introduction to the methods used in the combinatorics of pattern avoidance and pattern enumeration in compositions and words. It also presents various tools and approaches that are applicable to other areas of enumerative combinatorics.

After a historical perspective on research in the area, the text introduces techniques to solve recurrence relations, including iteration and generating functions. It then focuses on enumeration of basic statistics for compositions. The text goes on to present results on pattern avoidance for subword, subsequence, and generalized patterns in compositions and then applies these results to words. The authors also cover automata, the ECO method, generating trees, and asymptotic results via random compositions and complex analysis.

Highlighting both established and new results, this book explores numerous tools for enumerating patterns in compositions and words. It includes a comprehensive bibliography and incorporates the use of the computer algebra systems Maple™ and Mathematica®, as well as C++ to perform computations.

Autoren/Hrsg.

Heubach, Silvia

Mansour, Toufik

Fachgebiete

• Mathematik | Informatik EDV | Informatik Programmierung | Softwareentwicklung Algorithmen & Datenstrukturen

Inhaltsverzeichnis

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.

Über Autor(innen)

Silvia Heubach is a Professor and the Chair of the Department of Mathematics at the California State University, Los Angeles, where she received the Outstanding Professor Award in 1999/2000.
Toufik Mansour is an Associate Professor at the University of Haifa. The author or co-author of more than 60 papers, Professor Mansour’s general research interest is in discrete mathematics and its applications, with an emphasis on pattern avoidance problems.