Asymptotics and inequalities for partitions into squares

TL;DR

This paper proves that for large numbers, the counts of partitions into squares with even and odd parts are asymptotically equal, but differ depending on parity, confirming a recent conjecture.

Contribution

It establishes the asymptotic equality and parity-dependent difference in the number of partitions into squares, solving a conjecture by Bringmann and Mahlburg.

Findings

Asymptotic equality of partition counts for large n

Difference in counts depends on the parity of n

Confirmed the conjecture by Bringmann and Mahlburg

Abstract

In this paper we prove that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further show that, for $ n $ large enough, the two quantities are different and which of the two is bigger depends on the parity of $ n. $ This solves a recent conjecture formulated by Bringmann and Mahlburg (2012).