Proof of Xiong's conjectured refinement of Euler's partition theorem

William J. Keith

Published 2016-08-11Version 1

In a recent preprint, Xinhua Xiong conjectured a refinement of Euler's partition theorem that partitions with distinct parts are equinumerous with those into odd parts. His conjecture applies to all moduli and generalizes a result of Pak and Postnikov on partitions with all parts having the same residue with respect to a given modulus. This note bijectively proves Xiong's conjecture in its full generality.