Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds

In 1937 there appeared a paper that was to have a profound influence on the progress of combinatorial enumeration, both in its theoretical and applied aspects. Entitled Kombinatorische Anzahlbest­ immungen jUr Gruppen, Graphen und chemische Verbindungen, it was published in Acta Mathematica, Vol. 68, pp. 145 to 254. Its author, George Polya, was already a mathematician of considerable stature, well-known for outstanding work in many branches of mathematics, particularly analysis. The paper in Question was unusual in that it depended almost entirely on a single theorem -- the "Hauptsatz" of Section 4 -- a theorem which gave a method for solving a general type of enumera­ tion problem. On the face of it, this is not something that one would expect to run to over 100 pages. Yet the range of the applica­ tions of the theorem and of its ramifications was enormous, as Polya clearly showed. In the various sections of his paper he explored many applications to the enumeration of graphs, principally trees, and of chemical isomers, using his theorem to present a comprehen­ sive and unified treatment of problems which had previously been solved, if at all, only by ad hoc methods. In the final section he investigated the asymptotic properties of these enumerational results, bringing to bear his formidable insight as an analyst.

1 Groups.- 2 Graphs.- 3 Chemical Compounds.- 4 Asymptotic Evaluation of the Number of Combinations.- The Legacy of Pólya's Paper: Fifty Years of Pólya Theory.- References.