On the largest sizes of certain simultaneous core partitions with distinct parts

TL;DR

This paper determines the largest sizes of certain simultaneous core partitions with distinct parts, generalizing Amdeberhan's conjecture and showing the number of such partitions with the largest size is at most two.

Contribution

It extends the understanding of the largest sizes of $(t,mt\pm 1)$-core partitions with distinct parts, confirming a broader conjecture.

Findings

Largest sizes of $(t,mt\pm 1)$-core partitions with distinct parts are explicitly derived.

The number of such partitions with the largest size is at most two.

The results verify a generalization of Amdeberhan's conjecture.

Abstract

Motivated by Amdeberhan's conjecture on $(t,t+1)$-core partitions with distinct parts, various results on the numbers, the largest sizes and the average sizes of simultaneous core partitions with distinct parts were obtained by many mathematicians recently. In this paper, we derive the largest sizes of $(t,mt\pm 1)$-core partitions with distinct parts, which verifies a generalization of Amdeberhan's conjecture. We also prove that the numbers of such partitions with the largest sizes are at most $2$.