Combinatorics of Set Partitions

By Toufik Mansour

© 2012 – Chapman and Hall/CRC

516 pages | 38 B/W Illus.

Purchasing Options:
Hardback: 9781439863336
pub: 2012-07-27
US Dollars$122.95

About the Book

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.


"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

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

Generating Functions

Lagrange Inversion Formula

The Principle of Inclusion and Exclusion

Generating Trees

Preliminary Results on Set Partitions

Dobiński’s Formula

Different Representations

Subword Statistics on Set Partitions

Subword Patterns of Size Two: Rises, Levels and Descents

Peaks and Valleys

Subword Patterns: -Rises, -Levels, and -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

Major Index

Number of Crossings, Nestings and Alignments

Avoidance of Patterns in Set Partitions

History and Connections

Avoidance of Subsequence Patterns

Generalized Patterns

Partially Ordered Patterns

Multi Restrictions on Set Partitions

Avoiding a Pattern of Size Three and Another Pattern

Pattern Avoidance in Noncrossing Set Partitions

General Equivalences

Two Patterns of Size Four

Left Motzkin Numbers

Sequence A054391

Catalan and Generalized Catalan Numbers

Pell Numbers

Regular Set Partitions

Distance Restrictions



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((aa)n)

Wick’s Theorem and q-Normal Ordering

p-Normal Ordering

Noncrossing Normal Ordering




Exercises, Research Directions, and Open Problems appear at the end of each chapter.

About the Author

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.

About the Series

Discrete Mathematics and Its Applications

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Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Operating Systems / General
MATHEMATICS / Combinatorics