On 𝓁-Regular Partitions and Hickerson’s Identity

Based on two involutions and a bijection, we combinatorially determine the difference between the number of 𝓁-regular partitions of 𝑛 into an even number of parts and into an odd number of parts for all positive integers 𝑛 and 𝓁 > 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.