Combinatorics and Number Theory of Counting Sequences  book cover
1st Edition

Combinatorics and Number Theory of Counting Sequences

ISBN 9781138564855
Published August 20, 2019 by Chapman and Hall/CRC
498 Pages 10 B/W Illustrations

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Book Description

Combinatorics and Number Theory of Counting Sequences is an introduction to the theory of finite set partitions and to the enumeration of cycle decompositions of permutations.

The presentation prioritizes elementary enumerative proofs. Therefore, parts of the book are designed so that even those high school students and teachers who are interested in combinatorics can have the benefit of them. Still, the book collects vast, up-to-date information for many counting sequences (especially, related to set partitions and permutations), so it is a must-have piece for those mathematicians who do research on enumerative combinatorics.

In addition, the book contains number theoretical results on counting sequences of set partitions and permutations, so number theorists who would like to see nice applications of their area of interest in combinatorics will enjoy the book, too.


  • The Outlook sections at the end of each chapter guide the reader towards topics not covered in the book, and many of the Outlook items point towards new research problems.
  • An extensive bibliography and tables at the end make the book usable as a standard reference.
  • Citations to results which were scattered in the literature now become easy, because huge parts of the book (especially in parts II and III) appear in book form for the first time.

Table of Contents

I Counting sequences related to set partitions and permutations

Set partitions and permutation cycles.

Generating functions

The Bell polynomials

Unimodality, log concavity and log convexity

The Bernoulli and Cauchy numbers

Ordered partitions

Asymptotics and inequalities

II Generalizations of our counting sequences

Prohibiting elements from being together

Avoidance of big substructures

Prohibiting elements from being together

Avoidance of big substructures

Avoidance of small substructures

III Number theoretical properties


Congruences vial finite field methods

Diophantic results



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István Mező is a Hungarian mathematician. He obtained his PhD in 2010 at the University of Debrecen. He was working in this institute until 2014. After two years of Prometeo Professorship at the Escuela Politécnica Nacional (Quito, Ecuador) between 2012 and 2014 he moved to Nanjing, China, where he is now a full-time research professor.


This book provides an interesting introduction to combinatorics by employing number-theoretic techniques of counting sequences. The level of the presentation often seems elementary, as the author frequently throws out lagniappes suitable for high school students. The text unfolds in three parts. Part 1 covers set partitions, generating functions, Bell polynomials, log-concavity, log-convexity, Bernoulli and Cauchy numbers, ordered partitions, asymptotes, and related inequalities. Part 2 discusses generalizations of counting sequences in three chapters. The final part considers number theoretical properties, including congruences, by way of finite field methods and Diophantine results. Each chapter concludes with an "Outlook" section that gives suggestions about exploring additional topics not covered in the text. Mathematical proof is used throughout the exposition and tends to be "enumerative," again contributing to a sense that the author hopes to engage mathematical novices through this text. However, the more than 250 exercises included in the book are frequently challenging and always interesting. The bibliography comprises more than 600 entries. Anyone who can follow the text is likely to enjoy working through the book.
-D. P. Turner, Faulkner University