Abaci structures of (s, ms ± 1)-core partitions

We develop a geometric approach to the study of (s,ms–1)-core and (s, ms+1)- core partitions through the associated ms-abaci. This perspective yields new proofs for results of H. Xiong and A. Straub (originally proposed by T. Amdeberhan) on the enumeration of (s, s + 1) and (s, ms – 1)-core partitions with distinct parts. It also enumerates the (s, ms+1)-cores with distinct parts. Furthermore, we calculate the size of the (s, ms – 1, ms + 1)-core partition with the largest number of parts. Finally we enumerate self-conjugate core partitions with distinct parts. The central idea throughout is that the ms-abaci of largest (s, ms ± 1)-cores can be built up from s-abaci of (s, s ± 1)-cores in an elegant way.