  Using a combinatorial bijection with certain abaci diagrams, Nath and Sellers
have enumerated $(s, m s \pm 1)$-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 $(s, m s \pm r)$-core partitions. As an application of our
approach, we obtain the average and maximum number of parts of these core
partitions.
