Refined Counting of Core Partitions into $d$-Distinct Parts

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