Jehanne Dousse: Refinement of partition identities

Abstract

A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. A Rogers-Ramanujan type partition identity is a theorem stating that for all n, the number of partitions of n satisfying some difference conditions equals the number of partitions of n satisfying some congruence conditions. In 1993 Alladi and Gordon introduced the method of weighted words to find refinements of Schur's theorem and other partition identities. After explaining their original method which relies on q-series identities, we will present a new version using q-difference equations and recurrences. It allows one to prove refinements and generalisations of identities with intricate difference conditions for which the classical method is difficult to apply, such as identities of Primc and Siladic which arose in representation theory.