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.

part I|2 pages

Counting sequences related to set partitions and permutations

chapter Chapter 1|29 pages

Set partitions and permutation cycles

chapter Chapter 2|52 pages

Generating functions

chapter Chapter 3|20 pages

The Bell polynomials

chapter Chapter 4|17 pages

Unimodality, log-concavity, and log-convexity

chapter Chapter 5|18 pages

The Bernoulli and Cauchy numbers

chapter Chapter 6|25 pages

Ordered partitions

chapter Chapter 7|27 pages

Asymptotics and inequalities

part II|1 pages

Generalizations of our counting sequences

chapter 8|41 pages

Prohibiting elements from being together

chapter Chapter 9|27 pages

Avoidance of big substructures

chapter Chapter 10|28 pages

Avoidance of small substructures

part III|2 pages

Number theoretical properties

chapter Chapter 11|44 pages


chapter Chapter 12|22 pages

Congruences via finite field methods

chapter Chapter 13|18 pages

Diophantic results