1 - Decomposition of the integers as a direct sum of two subsets

Summary

Abstract. The study of identities of Rogers-Ramanuj an type forms an important part of the theory of partitions and q-series. These identities relate partitions whose parts satisfy certain difference conditions to partitions whose parts satisfy congruence conditions. Lie Algebras have provided a natural setting in which many such identities have arisen. In this paper a new technique called “the method of weighted words” is discussed and various applications illustrated. The method is particularly useful in obtaining generalisations and refinements of various Rogers-Ramanuj an type identities. In doing so, new companions to familiar identities emerge. Gordon and I introduced the method a few years ago to obtain generalisations and refinements of the celebrated 1926 partition theorem of Schur. The method has now been improved in collaboration with Andrews and Gordon thereby increasing its applicability. The improved method yielded a generalisation and a strong refinement of a recent partition conjecture of Capparelli which arose in a study of Lie Algebras. Another application is a refinement and generalisation of a deep partition theorem of Gollnitz. A unified approach to these partition identities is presented here by blending the ideas in four of my recent papers with Andrews and Gordon. Proofs of many of the results are given, but for those where the details are complicated, only the main ideas are sketched.

Introduction

Identities of Rogers-Ramanuj an type form an important part of the theory of partitions and q-series. Generally, one side of these identities is in the form of an infinite series, while the other side is an infinite product.