Proof of a limited version of Mao’s partition rank inequality using a theta function identity

Rupam Barman,Archit Pal Singh Sachdeva

doi:10.1007/s40993-016-0055-5

Abstract

Ramanujan’s congruence \(p(5k+4) \equiv 0 \pmod 5\) led Dyson (Eureka 8:10–15, 1944) to define a measure “rank”, and then conjectured that \(p(5k+4)\) partitions of \(5k+4\) could be divided into subclasses with equal cardinality to give a direct proof of Ramanujan’s congruence. The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer (Some properties of partitions 4:84–106, 1954), who proved Dyson’s conjecture. More recently, Mao (Number Theory 133:3678–3702, 2013) proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 and for \(M_2\)-rank differences for partitions with no repeated odd parts modulo 6 and 10 (Mao in Ramanujan J 37:391–419, 2015). Alwaise et al. (Ramanujan J. doi:10.1007/s11139-016-9789-x, 2016) proved four of Mao’s conjectured inequalities, while leaving three open. Here, we prove a limited version of one of the inequalities conjectured by Mao.

Highlights

Introduction and resultsA partition of a positive integer n is a way of writing n as a sum of positive integers, usually written in non-increasing order of the summands or parts of the partition
We denote the number of parts in the partition as n(λ) and the largest part as l(λ)
The celebrated Ramanujan congruences for the partition function begged for a combinatorial interpretation: p(5k + 4) ≡ 0, p(7k + 5) ≡ 0, p(11k + 6) ≡ 0

Summary

Introduction

Introduction and resultsA partition of a positive integer n is a way of writing n as a sum of positive integers, usually written in non-increasing order of the summands or parts of the partition. The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer (Some properties of partitions 4:84–106, 1954), who proved Dyson’s conjecture. Mao (Number Theory 133:3678–3702, 2013) proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 and for M2-rank differences for partitions with no repeated odd parts modulo 6 and 10 (Mao in Ramanujan J 37:391–419, 2015).

Results

Conclusion