Refined counting of core partitions into \(d\)-distinct parts (Q2227836)

Summary: Using a combinatorial bijection with certain abaci diagrams, \textit{R. Nath} and \textit{J. A. Sellers} [ibid., No. 1, Research Paper P1.5, 20 p. (2017; Zbl 1355.05039)] have enumerated \((s,ms\pm 1)\)-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 \((s,ms\pm r)\)-core partitions. As an application of our approach, we obtain the average and maximum number of parts of these core partitions.