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

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.