It is regrettable that a book, once published and on the way to starting a life of its own, can no longer bear witness to the painful choices that the author had to face in the course of his writing. There are choices that confront the writer of every book: who is the intended audience? who is to be proved wrong? who will be the most likely critic? Most of us have indulged in the idle practice of drafting tables of contents of books we know will never see the light of day. In some countries, some such particularly imaginative drafts have actually been sent to press (though they may not be included among the author's list of publications).

In mathematics, however, the burden of choice faced by the writer is so heavy as to turn off all but the most courageous. And of all mathematics, combinatorics is nowadays perhaps the hardest to write on, despite an eager audience that cuts across the party lines. Shall an isolated special result be granted a section of its own? Shall a fledgling new theory with as yet sparse applications be gingerly thrust in the middle of a chapter? Shall the author yield to one of the contrary temptations of recreational math at one end, and categorical rigor at the other? or to the highly rewarding lure of the algorithm?

Inclusion–Exclusion

Roughly speaking, a “sieve method” in enumerative combinatorics is a method for determining the cardinality of a set S that begins with a larger set and somehow subtracts off or cancels out the unwanted elements. Sieve methods have two basic variations: (1) We can first approximate our answer by an overcount, then subtract off an overcounted approximation to our original error, and so on, until after finitely many steps we have “converged” to the correct answer. This is the combinatorial essence of the Principle of Inclusion–Exclusion, to which this section and the next four are devoted. (2) The elements of the larger set can be weighted in a natural combinatorial way so that the unwanted elements cancel out, leaving only the original set S. We discuss this technique in Sections 2.5–2.7.

The Principle of Inclusion–Exclusion is one of the fundamental tools of enumerative combinatorics. Abstractly, the Principle of Inclusion–Exclusion amounts to nothing more than computing the inverse of a certain matrix. As such it is simply a minor result in linear algebra. The beauty of this principle lies not in the result itself, but rather in its wide applicability. We will give several examples of problems that can be solved by the Principle of Inclusion–Exclusion, some in a rather subtle way. First we state the principle in its purest form.

Enumerative combinatorics is concerned with counting the number of elements of a finite set S. This definition, as it stands, tells us little about the subject since virtually any mathematical problem can be cast in these terms. In a genuine enumerative problem, the elements of S will usually have a rather simple combinatorial definition and very little additional structure. It will be clear that S has many elements, and the main issue will be to count (or estimate) them all and not, for example, to find a particular element. Of course there are many variants of this basic problem that also belong to enumerative combinatorics and that will appear throughout this book.

There has been an explosive growth in combinatorics in recent years, including enumerative combinatorics. One important reason for this growth has been the fundamental role that combinatorics plays as a tool in computer science and related areas. A further reason has been the prodigious effort, inaugurated by G.-C. Rota around 1964, to bring coherence and unity to the discipline of combinatorics, particularly enumeration, and to incorporate it into the mainstream of contemporary mathematics. Enumerative combinatorics has been greatly elucidated by this effort, as has its role in such areas of mathematics as finite group theory, representation theory, commutative algebra, algebraic geometry, and algebraic topology.

This book has three intended audiences and serves three different purposes. First, it may be used as a graduate-level introduction to a fascinating area of mathematics.

How to Count

The basic problem of enumerative combinatorics is that of counting the number of elements of a finite set. Usually we are given an infinite class of finite sets Si where i ranges over some index set I (such as the nonnegative integers ℕ), and we wish to count the number f(i) of elements of each Si “simultaneously.” Immediate philosophical difficulties arise. What does it mean to “count” the number of elements of Si? There is no definitive answer to this question. Only through experience does one develop an idea of what is meant by a “determination” of a counting function f(i). The counting function f(i) can be given in several standard ways:

1. The most satisfactory form of f(i) is a completely explicit closed formula involving only well-known functions, and free from summation symbols. Only in rare cases will such a formula exist. As formulas for f(i) become more complicated, our willingness to accept them as “determinations” of f(i) decreases. Consider the following examples.

1.1.1. Example. For each n ∈ ℕ, let f(n) be the number of subsets of the set [n] = {1, 2, …, n}. Then f(n) = 2n, and no one will quarrel about this being a satisfactory formula for f(n).

1.1.2. Example. Suppose n men give their n hats to a hat-check person. Let f(n) be the number of ways that the hats can be given back to the men, each man receiving one hat, so that no man receives his own hat.

A square matrix is called doubly stochastic if all entries of the matrix are nonnegative and the sum of the elements in each row and each column is unity. Among the class of nonnegative matrices, stochastic matrices and doubly stochastic matrices have many remarkable properties. Whereas the properties of stochastic matrices are mainly spectral theoretic and are motivated by Markov chains, doubly stochastic matrices, besides sharing such properties, also have an interesting combinatorial structure. In this chapter we first consider the combinatorial properties of the polytope of doubly stochastic matrices. The Birkhoff—von Neumann Theorem, the Frobenius-König Theorem, and related results are proved. An extension of the Frobenius-König Theorem involving matrix rank is given. We then describe a probabilistic algorithm to find a positive diagonal in a nonnegative matrix. Such algorithms are of relatively recent origin. The next several sections focus on diagonal products and permanents of nonnegative as well as doubly stochastic matrices. The proof of the van der Waerden conjecture due to Egorychev is given. We also give an elementary alternative proof of the Alexandroff Inequality, which is along the lines of the proof of the van der Waerden conjecture due to Falikman. The last few sections are concerned with various problems in game theory, scheduling, and economics.

A variety of biological, statistical, and social science data come in the form of cross-classified tables of counts commonly known as contingency tables. Scaling the cell entries of such multidimensional matrices involves both mathematically and statistically well-posed problems of practical interest. In this chapter we first describe several situations where scaling can be useful. We then prove a very general theorem that demonstrates the existence of scaling factors. We also describe a natural scaling algorithm in the problem of scaling a nonnegative matrix to obtain prescribed row and column sums. In order to study the convergence properties of the algorithm it is convenient to work in terms of Hilbert's projective metric. Certain related concepts such as the contraction ratio of Birkhoff and the oscillation ratio of Hopf are introduced. In the last section we consider the problem of maximum likelihood estimation in contingency tables. This area of statistics, which forms part of discrete multivariate analysis, is of considerable interest to research workers at present.

Practical examples of scaling problems

Before we take up the mathematical problem of scaling, we illustrate some practical situations where scaling is useful.

Budget allocation problem

The Air Force, the Army, and the Navy have received their budget for the next fiscal year measured in some units to be allocated among technical, administrative, and research categories.

In this chapter we discuss certain combinatorial topics where positive semidefinite matrices and nonnegative matrices appear. In the first section we give a quick introduction to matroids and prove a result due to Rado, which includes Hall's Theorem on systems of distinct representatives as a special case. Then we discuss basic properties of the mixed discriminant, a function that allows a unified treatment of the theory of permanent of a nonnegative matrix and the determinant of a positive semidefinite matrix. The Alexandroff Inequality for mixed discriminants is proved and it is used to settle a special case of a conjecture of Mason. The next section deals with a topic in the area of spectra of graphs. Graphs whose adjacency matrices have Perron root less than 2 are characterized. These graphs correspond to the well-known Coxeter graphs (or Dynkin Diagrams). It is also shown that these graphs are precisely the ones giving rise to a finite Weyl group. The next section focuses on matrices over an algebraic structure called max algebra. As far as the eigenproblem is concerned, such matrices behave somewhat like nonnegative matrices. The last section deals with Boolean matrices. The main emphasis is on characterization of Boolean matrices that admit Moore-Penrose inverse.

Mathematical precision in describing models of macroeconomies became prominent after the advent of Leontief's monumental work on input-output analysis [Leontief (1941)]. In a totally independent setting, von Neumann's model of an expanding economy gave a new impetus to the mathematical approach to economic models [von Neumann (1937)]. Economists of earlier centuries were often too ambitious in their tasks of incorporating many complex economic issues into their models. When it came to the analysis of their models they resorted to many heuristic arguments. In contrast, modern mathematical economists believe in the analytical rigor of their arguments with no ambiguity in the final conclusions. However, they too have to pay a price for the same. Often, their drastically simplified mathematical models tend to avoid the serious economic issues, such as production and capital accumulation over several periods.

In the study of an economy, many of the variables such as prices, costs of production, rates of return, intensity of operations, etc. are clearly nonnegative. With the introduction of Leontief's input-output analysis, a good linear approximation to the functioning of an economy controlled by a few firms or the state has been achieved with great empirical success [see Miller and Blair (1985)]. The theory of nonnegative matrices and M-Matrices play an important role in the study of these models.

The Perron-Frobenius Theorem is central to the theory of nonnegative matrices. An irreducible nonnegative matrix can be viewed as the payoff matrix of a zero-sum, two-person game with positive value. A matrix game is said to be completely mixed if no row or column is dispensable for optimal play. In this chapter we first exploit the properties of completely mixed matrix games to prove the Perron-Frobenius Theorem. The next few sections deal with certain related topics such as M-matrices, the structure of reducible nonnegative matrices, primitive matrices, and polyhedral sets with a least element. We then describe the basic aspects of finite Markov chains. In the final section we prove the Perron-Frobenius Theorem for operators that leave the Lorentz cone invariant.

Irreducible nonnegative matrices

We work with real matrices throughout, unless stated otherwise. Let A = (aij) be an m × n matrix. We say that the matrix A is nonnegative and write A ≥ 0, if aij ≥ 0 for all i, j. If aij > 0 for all i, j, then the matrix A is called positive and we write A > 0. For matrices A, B, we say A ≥ B if A — B ≥ 0. Similar definitions and notation apply for vectors.