Refined counting of core partitions into $d$-distinct parts

Abstract

Using a combinatorial bijection with certain abaci diagrams, Nath and Sellers have enumerated (s, ms ± 1)-core partitions into distinct parts. We generalize their result in several directions by including the number of parts of these partitions, by considering d-distinct partitions, and by allowing more general (s, ms ± r)-core partitions. As an application of our approach, we obtain the average and maximum number of parts of these core partitions.