Core Partitions With d-Distinct Parts

TL;DR

This paper investigates the properties of (s,s+1)-core partitions with d-distinct parts, extending previous results and proposing a new conjecture for a broader class of core partitions.

Contribution

It extends existing work on core partitions with distinct parts by analyzing d-distinct parts and introduces a conjecture for (s,s+r)-core partitions.

Findings

Derived formulas for the number of such partitions

Determined the largest size of these partitions

Proposed a conjecture for (s,s+r)-core partitions with d-distinct parts

Abstract

In this paper, we study $(s,s+1)$-core partitions with $d$-distinct parts. We obtain results on the number and the largest size of such partitions, so we extend Xiong's paper in which the results are obtained about $(s,s+1)$-core partitions with distinct parts. Also, we propose a conjecture about $(s,s+r)$-core partitions with $d$-distinct parts for $1 \le r \le d$.