On $\ell$-regular partitions and Hickerson's identity

TL;DR

This paper establishes a comprehensive combinatorial framework to analyze the difference in counts of $oldsymbol{ extit{ ext{l}}}$-regular partitions with even and odd parts, extending recent results and providing a new proof of Hickerson's identity.

Contribution

It introduces two involutions and a bijection to fully determine the difference in partition counts for all $n$ and $oldsymbol{ extit{ ext{l}}}>1$, extending prior work.

Findings

Complete characterization of the difference between even and odd $ extit{ ext{l}}$-regular partitions.

A new combinatorial proof of Hickerson's identity.

Extension of previous results by Ballantine and Merca.

Abstract

Based on two involutions and a bijection, we completely determine the difference between the number of $\ell$-regular partitions of $n$ into an even number of parts and into an odd number of parts for all positive integers $n$ and $\ell>1$, which extends two recent results due to Ballantine and Merca. As an application, we provide a combinatorial proof of Hickerson's identity on the number of partitions into an even and odd number of parts.