A bijective proof of Amdeberhan's conjecture on the number of $(s, s+2)$-core partitions with distinct parts

TL;DR

This paper provides the first direct bijective proof of Amdeberhan's conjecture, establishing a one-to-one correspondence between certain core partitions and lattice paths, confirming the simple formula for their count.

Contribution

It presents the first bijective proof of the conjecture, linking core partitions with lattice paths, advancing combinatorial understanding.

Findings

Confirmed the number of $(s,s+2)$-core partitions with distinct parts is $2^{s-1}$ for odd $s$

Established a bijection between core partitions and lattice paths

Provided a new combinatorial proof method for the conjecture

Abstract

Amdeberhan conjectured that the number of $(s,s+2)$-core partitions with distinct parts for an odd integer $s$ is $2^{s-1}$. This conjecture was first proved by Yan, Qin, Jin and Zhou, then subsequently by Zaleski and Zeilberger. Since the formula for the number of such core partitions is so simple one can hope for a bijective proof. We give the first direct bijective proof of this fact by establishing a bijection between the set of $(s, s+2)$-core partitions with distinct parts and a set of lattice paths.