A generalisation of two partition theorems of Andrews

Jehanne Dousse

doi:10.46298/dmtcs.2529

Abstract

In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur’s celebrated partition identity (1926). Andrews’ two generalisations of Schur’s theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In this paper we generalise both of Andrews’ theorems to overpartitions. The proofs use a new technique which consists in going back and forth from $q$-difference equations on generating functions to recurrence equations on their coefficients. En 1968 et 1969, Andrews a prouvé deux identités de partitions du type Rogers-Ramanujan qui généralisent le célèbre théorème de Schur (1926). Ces deux généralisations sont devenues deux des théorèmes les plus importants de la théorie des partitions, avec des applications en combinatoire, en théorie des représentations et en algèbre quantique. Dans ce papier, nous généralisons les deux théorèmes de Andrews aux surpartitions. Les preuves utilisent une nouvelle technique qui consiste à faire des allers-retours entre équations aux $q$-différences sur les séries génératrices et équations de récurrence sur leurs coefficients.

Highlights

A partition of n is a non-increasing sequence of natural numbers whose sum is n
Theorem 1.5 Let F (−AN ; k, n) denote the number of overpartitions of n into parts taken from −AN, having k non-overlined parts
Let G(−AN ; k, n) denote the number of overpartitions of n into parts taken from −AN of the form n = λ1 + · · · + λs, having k non-overlined parts, such that λi − λi+1 ≥ N w βN (−λi) − 1 + χ(λi+1) + v(βN (−λi)) − βN (−λi), λs ≥ N (w(βN (−λs)) − 1), where χ(λi+1) = 1 if λi+1 is overlined and 0 otherwise

Summary

Introduction

A partition of n is a non-increasing sequence of natural numbers whose sum is n. Theorem 1.3 (Andrews) Let F (−AN ; n) denote the number of partitions of n into distinct parts taken from −AN. Theorem 1.5 Let F (−AN ; k, n) denote the number of overpartitions of n into parts taken from −AN , having k non-overlined parts. Let G(−AN ; k, n) denote the number of overpartitions of n into parts taken from −AN of the form n = λ1 + · · · + λs, having k non-overlined parts, such that λi − λi+1 ≥ N w βN (−λi) − 1 + χ(λi+1) + v(βN (−λi)) − βN (−λi), λs ≥ N (w(βN (−λs)) − 1), where χ(λi+1) = 1 if λi+1 is overlined and 0 otherwise. Theorem 1.6 Let D(AN ; k, n) denote the number of overpartitions of n into parts taken from AN , having k non-overlined parts.

The recurrence equation

Evaluating lim u by induction

Ideas for future research