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

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 $2$.

arXiv comment9 pages

References in corpus (4)

• Partitions with prescribed hooksets
• Explicit expressions for the moments of the size of an (s,s+1)-core partition with distinct parts
• Core partitions with distinct parts
• Explicit (Polynomial!) Expressions for the Expectation, Variance and Higher Moments of the Size of a (2n + 1, 2n + 3)-core partition with Distinct Parts