Early developments

Certain types of combinatorial problems have attracted the attention of mathematicians since early times. Magic squares, for example, which are squarearrays of numbers with the property that the rows, columns, and diagonals add up to the same number, occur in theI Ching, a Chinese book dating back to the 12th centurybc. Thebinomial coefficients, or integer coefficients in the expansion of (a +b)ⁿ, were known to the 12th-century Indian mathematicianBhāskara, who in hisLīlāvatī (“The Graceful”), dedicated to a beautiful woman, gave the rules for calculating them together with illustrative examples. “Pascal’s triangle,” a triangular array of binomial coefficients, had been taught by the 13th-century Persian philosopher Naṣīr ad-Dīn aḷ-Ṭūsī.

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In the West, combinatorics may be considered to begin in the 17th century withBlaise Pascal andPierre de Fermat, both of France, who discovered many classical combinatorial results in connection with the development of the theory of probability. The term combinatorial was first used in the modern mathematical sense by the German philosopher and mathematicianGottfried Wilhelm Leibniz in hisDissertatio de Arte Combinatoria (“Dissertation Concerning the Combinational Arts”). He foresaw the applications of this newdiscipline to the whole range of the sciences. The Swiss mathematicianLeonhard Euler was finally responsible for the development of a school of authentic combinatorial mathematics beginning in the 18th century. He became the father of graph theory when he settled theKönigsberg bridge problem, and his famous conjecture on Latin squares was not resolved until 1959.

In England,Arthur Cayley, near the end of the 19th century, made important contributions to enumerative graph theory, andJames Joseph Sylvester discovered many combinatorial results. The British mathematicianGeorge Boole at about the same time used combinatorial methods in connection with the development ofsymbolic logic, and the combinatorial ideas and methods ofHenri Poincaré, which developed in the early part of the 20th century in connection with the problem ofn bodies, have led to the discipline oftopology, which occupies the centre of the stage of mathematics. Many combinatorial problems were posed during the 19th century as purely recreational problems and are identified by such names as “the problem of eight queens” and “the Kirkman school girl problem.” On the other hand, the study of triple systems begun by Thomas P. Kirkman in 1847 and pursued byJakob Steiner, a Swiss-born German mathematician, in the 1850s was the beginning of the theory ofdesign. Among the earliest books devoted exclusively to combinatorics are the German mathematician Eugen Netto’sLehrbuch der Combinatorik (1901; “Textbook of Combinatorics”) and the British mathematician Percy Alexander MacMahon’sCombinatory Analysis (1915–16), which provide a view of combinatorial theory as it existed before 1920.

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Combinatorics during the 20th century

Many factors have contributed to the quickeningpace of development of combinatorial theory since 1920. One of these was the development of the statistical theory of the design of experiments by the English statisticians Ronald Fisher and Frank Yates, which has given rise to many problems of combinatorial interest; the methods initially developed to solve them have found applications in such fields as coding theory. Information theory, which arose around midcentury, has also become a rich source of combinatorial problems of a quite new type.

Another source of the revival of interest in combinatorics isgraph theory, the importance of which lies in the fact that graphs can serve as abstract models for many different kinds of schemes of relations among sets of objects. Its applications extend tooperations research, chemistry,statistical mechanics,theoretical physics, and socioeconomic problems. The theory of transportation networks can be regarded as a chapter of the theory of directed graphs. One of the most challenging theoretical problems, the four-colour problem (see below) belongs to the domain of graph theory. It has also applications to such other branches of mathematics asgroup theory.

The development of computer technology in the second half of the 20th century is a main cause of the interest in finite mathematics in general and combinatorial theory in particular. Combinatorial problems arise not only innumerical analysis but also in the design of computer systems and in the application of computers to such problems as those of information storage and retrieval.

Statistical mechanics is one of the oldest and most productive sources of combinatorial problems. Much important combinatorial work has been done by applied mathematicians and physicists since the mid-20th century—for example, the work on Ising models (see belowThe Ising problem).

In pure mathematics, combinatorial methods have been used with advantage in suchdiverse fields as probability,algebra (finite groups and fields,matrix and lattice theory),number theory (difference sets),set theory (Sperner’s theorem), and mathematical logic (Ramsey’s theorem).

In contrast to the wide range of combinatorial problems and the multiplicity of methods that have been devised to deal with them stands the lack of a central unifying theory. Unifying principles and cross connections, however, have begun to appear in various areas of combinatorial theory. The search for anunderlying pattern that may indicate in some way how the diverse parts of combinatorics are interwoven is a challenge that faces mathematicians in the last quarter of the 20th century.