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

TL;DR

This paper generalizes the enumeration of core partitions with distinct parts by including the number of parts, considering $d$-distinct partitions, and analyzing more general $(s, m s \pm r)$-core partitions, providing new combinatorial insights.

Contribution

It extends previous enumeration results to broader classes of core partitions, incorporating the number of parts and generalizing to $d$-distinct and $(s, m s \pm r)$-core partitions.

Findings

Derived formulas for the average number of parts.

Determined the maximum number of parts in these partitions.

Established combinatorial bijections with abaci diagrams.

Abstract

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.