Combinatorics: First Steps


Mykola Perestyuk, PhD (Author) – Professor, Taras Shevchenko National University of Kyiv, Ukraine
Volodymyr Vyshenskyi, PhD (Author)

Series: Computational Mathematics and Analysis
BISAC: MAT036000

The main goal of our book is to provide easy access to the basic principles and methods that combinatorial calculations are based upon. The rule of product, the identity principle, recurrence relations and inclusion-exclusion principle are the most important of the above. Significant parts of the book are devoted to classical combinatorial structures, such as: ordering (permutations), tuples, and subsets (combinations). A great deal of attention is paid to the properties of binomial coefficients, and in particular, to model proofs of combinatorial identities. Problems concerning some exact combinatorial configurations such as paths in a square, polygonal chains constructed with chords of a circle, trees (undirected graphs with no cycles) etc. are included too. All chapters contain a considerable number of exercises of various complexity, from easy training tasks to complex problems which require decent persistence and skill from the one who dares to solve them.

If one aims to passively familiarize oneself with the subject, methods and the most necessary facts of combinatorics, then it may suffice to limit one’s study to the main text omitting the exercise part of the book. However, for those who want to immerse themselves in combinatorial problems and to gain skills of active research in that field, the exercise section is rather important.

The authors hope that the book will be helpful for several categories of readers. University teachers and professors of mathematics may find somewhat unusual coverage of certain matters and exercises which can be readily applied in their professional work. We believe that certain series of problems may serve as a base for serious creative works and essays. This especially refers to students at pedagogical universities and colleges who need to prepare themselves to the teaching of the basics of combinatorics, mainly building on arithmetic and geometry. Most of the exercises of the book are of this very origin.


Table of Contents


Chapter 1. Elementary Enumerations of Combinations

Chapter 2. Basic Concepts of Set Theory

Chapter 3. Basic Combinatorial Structures

Chapter 4. Paths in a Rectangle

Chapter 5. Inclusion-exclusion Principle

Chapter 6. Trajectories Inside a Circle

Chapter 7. Trees



“This book is a very good introduction to combinatorics as a first step to know about the subject which is a basic for such areas as theory probability and statistics, among others. For example, Basic Concepts of Set Theory (Chapter 2) and Inclusion-Exclusion Principle (Chapter 5) are good examples of analogue between operations with sets and events in probability. Also, Trees (Chapter 7) have applications in financial mathematics, as we recall, binomial trees, as one of many examples. The book contains a great deal of problems (e.g., 160 (!), and it’s only in Chapter 1) with answers, examples and exercises. I would recommend it to university instructors who teach combinatorics, to pedagogical universities and colleges students, and also to high school students and first year university students. Moreover, as a background, mathematical courses from regular school will suffice to read and study the book. It is a really the first step in combinatorics to proceed to many more steps further in students’ journey through the beauty of combinatorics. Finally, I am proud to say, that one of the authors, namely, Volodymyr Vyshenskyi, was one of my best instructors in algebra at Kyiv State University back to 70th , when I was there as a freshman student. His pedagogical and teaching skills were a legend among young university’s students. All in all, do not be afraid to make the first step in anything, including combinatorics: it will pay-off when you make more steps and affords later.”  – Anatoliy Swishchuk, Professor, Applied Mathematics, Universiry of Calgary, Alberta, Canada

The book “Combinatorics: First Steps” written by two very experienced mathematicians, Mykola Perestyuk and Volodymyr Vyshenskyi, is a gem, which will satisfy the needs of all kinds of combinatorics lovers, from high school students to college professors. It has plenty of examples and solved problems that make the digesting of the material easy, though the phrase “First steps” in the title should not discourage know-it-alls from acquiring the book. It can also serve as a wonderful source of all kinds of combinatorial problems for extracurricular activities in mathematics.” – Vyacheslav Futorny, Professor, Departamento de Matematica, Universidade de Sao Paulo, Sao Paulo, Brazil

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