I have gone through the whole book as the topics are so interesting and written and explained in a perfect way.
Every chapter is nicely and properly explained.
From the author’s preface:
“This book is, on one hand, an introduction to the theory of finite set partitions and to the enumeration of cycle decompositions of permutations; on the other hand, it is an extensive collection of references to newer works, and also, it contains number theoretical results on counting sequences of set partitions and permutations.
During writing, I tried to be as “combinatorial” as possible, prioritizing “enumerative” proofs wherever it is feasible. Still, there is a large number of analytic tools presented in the text, like generating functions (as there is no modern combinatorics without these), asymptotics (which provide valuable large-scale information on the studied sequences), and a bit of Riordan arrays, Hankel determinants, and orthogonal polynomials, to mention a few.
I suppose that some (or many) of the readers of this book are new to the topic. Therefore, I intended to be as elementary as possible so that a part of the text is available even on a high school level. Enthusiasts such as teachers and students can pick several identities from the book (especially Chapters 1, 6, 8–9), and can try to provide proof for them with a high probability of success. The self-training reader will find the exercises useful at the end of the chapters.
The book is useful for researchers also, as it collects vast information for many counting sequences (especially related to set partitions and permutations). Some parts of the text (mainly in Chapters 8–9, 13) appear for the first time in this book or in a unified way.
At the end of many chapters, there is an Outlook which guides the reader towards new topics and discusses the available results in the literature which do not fit in this book because of page limitations or because of the complexity of the tools and proofs. Also, the Outlooks contain results with respect to generalizations or some more particular subjects. These Outlooks, together with an extensive Bibliography (616 entries) and Tables at the end, make the book usable as a standard reference for combinatorialists working in the field.
An extremely useful and widely used tool in the everyday work of those who deal with integer sequences is the On-line Encyclopedia of Integer Sequences (OEIS). We shall occasionally cite entries from this encyclopedia via their IDs.
During the past centuries, a very large number of scientists have contributed to this area. For non-contemporary authors, we tried to give a short footnote with some very basic information so that the reader might guide himself or herself when the particular discovery and contribution was made.
Chapter/Section headings:
I. Counting sequences related to set partitions and permutations.
1 Set partitions and permutation cycles; 2 Generating functions. 3 The Bell polynomials. 4 Unimodality, log-concavity, and log-convexity. 5 The Bernoulli and Cauchy numbers. 6 Ordered partitions. 7 Asymptotics and inequalities.
II Generalizations of our counting sequences.
8 Prohibiting elements from being together. 9 Avoidance of big substructures. 10 Avoidance of small substructures.
III Number theoretical properties.
11 Congruences. 12 Congruences via finite field methods. 13 Diophantic results.
Appendix: Basic combinatorial notions. The polynomial theorem. The Lambert W function.”
For the large amount of interesting subsections and the huge list of references see the publisher’s preview.