Focusing on a very active area of mathematical research in the last decade, Combinatorics of Set Partitions presents methods used in the combinatorics of pattern avoidance and pattern enumeration in set partitions. Designed for students and researchers in discrete mathematics, the book is a one-stop reference on the results and research activities of set partitions from 1500 A.D. to today.
Each chapter gives historical perspectives and contrasts different approaches, including generating functions, kernel method, block decomposition method, generating tree, and Wilf equivalences. Methods and definitions are illustrated with worked examples and Maple™ code. End-of-chapter problems often draw on data from published papers and the author’s extensive research in this field. The text also explores research directions that extend the results discussed. C++ programs and output tables are listed in the appendices and available for download on the author’s web page.
Table of Contents
Historical Overview and Earliest Results
Timeline of Research for Set Partitions
A More Detailed Book
Basic Tools of the Book
Solving Recurrence Relations
Lagrange Inversion Formula
The Principle of Inclusion and Exclusion
Preliminary Results on Set Partitions
Subword Statistics on Set Partitions
Subword Patterns of Size Two: Rises, Levels and Descents
Peaks and Valleys
Subword Patterns: ℓ-Rises, ℓ-Levels, and ℓ-Descents
Families of Subword Patterns
Patterns of Size Three
Nonsubword Statistics on Set Partitions
Statistics and Block Representation
Statistics and Canonical and Rook Representations
Records and Weak Records
Number of Positions between Adjacent Occurrences of a Letter
The Internal Statistic
Statistics and Generalized Patterns
Number of Crossings, Nestings and Alignments
Avoidance of Patterns in Set Partitions
History and Connections
Avoidance of Subsequence Patterns
Partially Ordered Patterns
Multi Restrictions on Set Partitions
Avoiding a Pattern of Size Three and Another Pattern
Pattern Avoidance in Noncrossing Set Partitions
Two Patterns of Size Four
Left Motzkin Numbers
Catalan and Generalized Catalan Numbers
Regular Set Partitions
Asymptotics and Random Set Partition
Tools from Probability Theory
Tools from Complex Analysis
Set Partitions as Geometric Words
Asymptotics for Set Partitions
Gray Codes, Loopless Algorithms and Set Partitions
Gray Code and Loopless Algorithms
Gray Codes for Pn
Loopless Algorithm for Generating Pn
Set Partitions and Normal Ordering
Linear Representation and N((a†a)n)
Wick’s Theorem and q-Normal Ordering
Noncrossing Normal Ordering
Exercises, Research Directions, and Open Problems appear at the end of each chapter.
Toufik Mansour is a professor in the Department of Mathematics at the University of Haifa. Dr. Mansour has authored/co-authored more than 200 papers and is a reviewer for many journals, including Advances in Applied Mathematics, Discrete Mathematics, Discrete Applied Mathematics, European Journal of Combinatorics, and the Journal of Combinatorial Theory Series A. His research focuses on pattern avoidance in permutations, colored permutations, set partitions, words, and compositions.
"Containing 375 references, this book works best as an encyclopedia for researchers working in this topic. But it can also be effectively used by students who are interested in the combinatorics of set partitions and want to get a hand on researching them. … a very useful reference for researchers of the enumerative side of set partitions."
—Acta Sci. Math. (Szeged), 80, 2014
"… a comprehensive account of the history and current research in the combinatorics of pattern enumeration and pattern avoidance in set partitions. … While it is aimed primarily at advanced undergraduate and graduate students in discrete mathematics with a focus on set partitions, its extensive bibliography, with 375 entries, and the variety of constructions and approaches used in the text make it a valuable reference for researchers in this field."
—Ricardo Mamede, Zentralblatt MATH 1261
"… a comprehensive account of recent and current research on the pattern-related aspects of set partitions."
—David Callan, Mathematical Reviews, April 2013