The Legacy of Pólya’s Paper: Fifty Years of Pólya Theory

Abstract

Pólya’s paper, translated here for the first time, was a landmark in the history of combinatorial analysis. It presented to mathematicians a unified technique for solving a wide class of combinatorial problems — a technique which is summarized in Pólya’s main theorem, the “Hauptsatz” of Section 16 of his paper, which will here be referred to as “Pólya’s Theorem”. This theorem can be explained and expounded in many different ways, and at many different levels, ranging from the down-to-earth to highly abstract. It will be convenient for future reference to review the essentials of Pólya’s Theorem, and to this end I offer the following, rather mundane, way of looking at the type of problem to which the theorem applies and the way that it provides a solution.