With advances in cybernetics and closely related divisions of science, discrete mathematics has found increasing importance as a tool for the investigation of various models of functioning of technical devices and discretely operating systems.

A significant place in discrete mathematics is occupied by combinatorial methods whose applications can help in solving the problems of the existence and construction of arrangements of elements according to certain rules, and in the estimation of the number of such arrangements. Each arrangement determines a configuration which can be considered as a mapping of one set onto another with some restrictions posed by a particular problem. If the restrictions are complicated, then we are faced with the problem of determining conditions of existence and suggesting methods of construction of such configurations.

In Chapter 1 such questions are considered for block-designs and Latin squares. The results presented in the chapter are intended to provide an insight into the typical problems of this area of combinatorial mathematics.

Chapter 2 is devoted to transversals, usually referred to as systems of distinct representatives of a family of sets. Permanents are the main tools for calculating the number of transversals of a family of sets. In this chapter methods of calculating the values of permanents are considered. These calculations meet with more difficulties than do calculations of determinants, those objects which in respect of many other properties are close to permanents.

We have pointed out the important role of formalizing the notion of the indistinguishability of elements in enumerative combinatorial problems. A fundamental contribution to the development of the methods of such formalization was made by Pólya in his famous paper (Pólya, 1937). The ideas in this paper were developed by de Bruijn (Beckenbach, 1964) and other mathematicians. It should be noted that, prior to the paper (Pólya, 1937), a similar method had been suggested by Redfield in (Redfield, 1927).

The essence of the method of solving enumerative problems, referred to as Pólya's enumeration theory, can be described as follows. There is a set Y of elements such that each element possesses a characteristic or weight which takes values from a ring. The distribution of the elements of the set over the weights is determined by a generating function F, usually of several variables. The set of configurations of the form f : X → Y, where X is a linearly ordered set, is considered. A characteristic or weight is also assigned to each configuration of the set. A group A of permutations generates an equivalence relation on the set of all configurations which determines the notion of the distinguishability of configurations. The non-equivalent configurations with given characteristics or weights are enumerated by a generating function Φ. The main theorem of Pólya's enumeration theory gives an expression of the generating function F in terms of the generating function F using some polynomial Z of several variables called the cycle index.

In this chapter we consider enumerative problems of graph theory, that is, problems arising when counting graphs with specific properties. We also consider similar problems about mappings of finite sets with various constraints. Particular attention is given to mappings of bounded height h, whose cycle lengths are the elements of a given sequence A. For h = 0 these mappings become substitutions whose cycle lengths belong to a given sequence A.

The method of generating functions can be effectively applied to these problems. As a result, we obtain either explicit formulae or some expressions for generating functions which allow us to find the asymptotic expressions of the corresponding coefficients, for instance by using the saddle point method. The results of the application of the saddle point method to the derivation of such asymptotic expressions are not given here. They can be found, for example, in papers (Sachkov, 1972; Sachkov, 1971b; Sachkov, 1971a).

The generating functions for graphs

Basic definitions

Let us formulate the basic definitions about the graphs; we follow the most common terminology of graph theory (Berge, 1958; Ore, 1962; Harary, 1969). A graph Γ = Γ(X, W) consists of a set X containing n ≥ 1 elements called vertices and of a set W of unordered pairs of vertices called edges. Usually a graph is geometrically represented on the plane by points corresponding to the vertices, and lines corresponding to the edges which join pairs of vertices from W; the intersections of the lines at points that differ from the vertices are not taken into account.

Beginning with Leibniz, a number of mathematicians have pointed out the fundamental importance of the distinguishability of objects in combinatorial problems, A rigorous definition of this notion requires certain formal notions which allow us to express the indistinguishability of objects in mathematical terms.

Consider as an example the formalization of the notion of indistinguishability in the well-known combinatorial scheme of allocation of particles to cells, usually referred as the urn model. There are m particles and n cells of unlimited capacity. The first problem consists of finding the number of possible allocations of the particles provided the allocations satisfy some restrictions. Obviously, in combinatorial problems of such a kind it is necessary to know which allocations are distinct and which ones are indistinguishable. First of all, the answer depends on whether the cells and the particles are distinguishable.

In Chapter 1 for such a formalization of combinatorial models we used the notion of a mapping of a finite set X into a set Y and introduced the notion of a configuration. With that approach the allocations of m particles into n cells were put into a one-to-one correspondence with configurations φ : X → Y, | X | = m, | Y | = n, and the distinguishability of allocations was defined in terms of the corresponding configurations. In this chapter we use another approach to the formalization of combinatorial schemes of the type considered, which is also based on the notion of a mapping. Let us illustrate the essence of the approach using the urn model again.

As the basic model we consider the scheme of allocation of m distinct particles to n different cells.

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}.