On -regular partitions and Hickerson's identity

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