On $(2k+1, 2k+3)$-core partitions with distinct parts

TL;DR

This paper investigates the enumeration of specific core partitions with distinct parts, confirming conjectures about their quantity and maximum size.

Contribution

It provides exact counts and size bounds for $(2k+1, 2k+3)$-core partitions with distinct parts, validating prior conjectures.

Findings

Number of such partitions is explicitly determined.

Largest size of these partitions is established.

Two conjectures by Straub are confirmed.

Abstract

In this paper, we are mainly concerned with the enumeration of $(2k+1, 2k+3)$-core partitions with distinct parts. We derive the number and the largest size of such partitions, confirming two conjectures posed by Straub.