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 is graph 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.
There is again an important class of theorems that guarantee the existence of certain choices under appropriate hypotheses.Besides their intrinsic interest, these theorems may be used as existence theorems in various combinatorial problems. As an example, a function Certain types of combinatorial problems have attracted the attention of mathematicians since early times.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 below The Ising problem).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 in numerical 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.Such a diagram for 14 = 5 3 3 2 1 is shown in and ϕ is the Euler function .Though the problem of the necklaces appears to be frivolous, the formula given above can be used to solve a difficult problem in the theory of Lie algebras, of some importance in modern physics.In the West, combinatorics may be considered to begin in the 17th century with Blaise Pascal and Pierre 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 mathematician Gottfried Wilhelm Leibniz in his Leonhard Euler was finally responsible for the development of a school of authentic combinatorial mathematics beginning in the 18th century.Kirkman in 1847 and pursued by Jakob Steiner, a Swiss-born German mathematician, in the 1850s was the beginning of the theory of Many factors have contributed to the quickening pace of development of combinatorial theory since 1920.