A few additions, which did not change the structure and level of presentation, have been included in the preparation of this English translation. Chapter 1 is supplemented by Section 1.8. In Chapter 2 insertions have been added to Subsection 2.1.3; this chapter is supplemented by Subsection 2.2.4 and Section 2.7; Section 2.6 is supplemented by Subsections 2.6.2 and 2.6.6. In Chapter 3, Sections 3.1, 3.3 and 3.6 are supplemented, and a new section, Section 3.4, is introduced. In Chapter 5, Subsection 5.2.6, item (g) in Section 5.4 and Section 5.9 have been added. Subsection 6.1.3 has been added to Chapter 6. In the main, the option of using certain parts of the book as a textbook, as well as the independence of the chapters, is preserved.

After the Russian edition of the book had appeared, a number of significant monographs on combinatorics and closely related problems were published. These books and some papers have been included in the Bibliography.

I am grateful to Professor B. Bollobás for his kind suggestion to Cambridge University Press to publish this book in English, and to Professor V. F. Kolchin for useful discussions during the translation.

This book is addressed to those who are interested in combinatorial methods of discrete mathematics and their applications. A major part of the book can be used as a textbook on combinatorial analysis for students specializing in mathematics. The remaining part is suitable for use in special lectures and seminars for the advanced study of combinatorics. Those parts which are not intended for teaching include Sections 2.3, 3.6, 3.7, 5.3, 5.6, 5.8, 6.3, 6.4 and Subsection 5.1.3 where the material contains either special questions concerning applications of combinatorial methods or rather cumbersome derivations of asymptotic formulae. Of course, a course of studies in discrete mathematics can be biased towards asymptotic methods, where the selection of material can be different and where the above-mentioned sections become basic.

Some knowledge of algebra and set theory, summarized in Section 1.1, is assumed. To understand the derivations of asymptotic formulae, the reader must be familiar with those results of complex analysis usually included in standard courses for students specializing in mathematics.

For the convenience of those readers who are interested in the separate questions contained in the book, I have attempted to make the presentation of each chapter self-contained and, for the most part, independent of the other chapters.

As is usual, I acknowledge those authors whose results are presented in the book and provide the corresponding citation. The list of references is given at the end of the book.

The method of citation is unified. Citations of theorems, lemmas, corollaries, formulae, etc., include the chapter number, section number and own number within the section. For example, Theorem 1.2.3 is theorem number 3 in Section 2 of Chapter 1.

The generating functions considered in this chapter are important instruments for solving the so-called enumerative problems in combinatorial analysis. Enumerative problems arise if we need to be explicit about the number of ways of choosing particular elements from a finite set. The application of generating functions in this situation consists of establishing a correspondence between the elements of the set and the terms of the products of some series; the solution of enumerative problems is reduced, in fact, to finding a suitable method for the multiplication of these series. Under these conditions, the convergence of the series is not necessary, and it is natural to use a formal power series, assuming that the operations on them are properly defined. The formal power series, generally speaking, of several variables, are called the generating functions.

Note that the application of generating functions to the solution of enumerative problems connected with establishing a correspondence between the elements of a set and the terms of formal series is an intermediate problem of combinatorial analysis. To solve the main problem (consisting of the derivation of expressions for the number of elements in a set depending on the parameters determining this set) it is appropriate to consider the corresponding power series as convergent in some domain of variation of a real or complex variable. Inside such domains the power series determine analytic functions whose properties are well known in classical analysis. We do not introduce a new terminology for the analytic functions applied to the solution of enumerative problems, but, rather, we call them the generating functions also.

The first section of this chapter is of an introductory nature and presents a summary of the main notions and results from the set theory and algebra which will be used in the book. In the sections that follow we consider various combinatorial configurations which may be introduced on the basis of the general notion of a configuration given in terms of mappings of sets. As examples of combinatorial configurations we consider Latin squares, orthogonal Latin squares, block designs and finite projective planes.

Notions of set theory and algebra

Boolean operations on sets

A set is a collection of elements of abstract nature, objects or notions, united by some common property. Along with the word “set” we sometimes use equivalent words such as “collection”, “family”, etc. A set consists of elements, and the formula x ∈ X means that the element x belongs to the set X; otherwise we write x ∉ X. If for each x ∈ X the inclusion x ∈ Y holds, then we say that X is a subset of Y and write X ⊆ Y. Sets X and Y are equal if X ⊆ y and Y ⊆ X. We say that a set X is a proper subset of Y and write X ⊂ Y if X ⊆ y and X ≠ Y. Any set contains, as a subset, the empty set denoted by ø.

A wide range of the so-called combinatorial problems of choice can be reduced to finding a system of distinct representatives for a given family of subsets of a set. In what follows, such a system will be called a transversal. We prefer this term because it is short and so has an advantage over the corresponding, more detailed, conventional term. The main questions considered in this chapter are related to the existence and number of transversals. The basis for the answers to the first series of questions is an existence theorem due to P. Hall, and various applications of it. To determine the number of transversals, a notion of a permanent is used which is a modification of the well-known notion of a determinant playing an important role in algebra.

The theorem of P. Hall is the basis for the proofs of the theorem of M. Hall on the existence of Latin squares and rectangles and Birkhoff's theorem on the representation of a stochastic matrix as a weighted sum of permutation matrices. Birkhoff's theorem is connected with a number of assertions about the decomposition of probabilistic automata and Markov chains with doubly stochastic matrices of transition probabilities.

Transversals

The main theorems

Let X be an arbitrary, generally speaking, infinite set; let X1, …, Xn be a family of subsets of X containing, in general, infinite subsets. Note that the equalities Xi = Xj for i ≠ j are allowed. We denote this family by (Xi: i ∈ I), where I = {1, …, n}.