On the largest size of $(t,t+1,..., t+p)$-core partitions

TL;DR

This paper proves Amdeberhan's conjecture on the maximum size of certain core partitions and characterizes the number of such partitions with the largest size, extending to a general case.

Contribution

It confirms the conjecture for three-core partitions and generalizes the results to multiple-core partitions, providing exact maximum sizes and counts.

Findings

Confirmed Amdeberhan's conjecture for (t,t+1,t+2)-core partitions.

Determined the maximum size and count of largest partitions based on parity of t.

Extended results to (t,t+1,..., t+p)-core partitions.

Abstract

In this paper we prove that Amdeberhan's conjecture on the largest size of $(t, t+1, t+2)$-core partitions is true. We also show that the number of $(t, t + 1, t + 2)$-core partitions with the largest size is $1$ or $2$ based on the parity of $t$. More generally, the largest size of $(t,t+1,..., t+p)$-core partitions and the number of such partitions with the largest size are determined.