Combinatorics and Number Theory of Counting Sequences

Combinatorialists are seldom aware of number theoretical tools, and number theorists rarely aware of possible combinatorial applications. This book is accessible for both of the groups. The first part introduces important counting sequences. The second part shows how these sequences can be generalized to study new combinatorial problems

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. Features

• 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.

  Autorentext

István Mez**** 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.

Inhalt

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

Congurences

Congruences vial finite field methods

Diophantic results

Appendix