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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 perspectiveyields new proofs for results of H. Xiong and A. Straub (originally proposed byT. Amdeberhan) on the enumeration of $(s, s+1)$ and $(s,ms-1)$-core partitionswith distinct parts. It also enumerates the $(s, ms+1)$-cores with distinctparts. Furthermore, we calculate the weight of the $(s, ms-1,ms+1)$-corepartition with the largest number of parts. Finally we use 2-core partitions toenumerate self-conjugate core partitions with distinct parts. The central ideais that the $ms$-abaci of maximal $(s,ms\pm1)$-cores can be built up from$s$-abaci of $(s,s\pm 1)$-cores in an elegant way.

Abaci structures of $(s,ms\pm1)$-core partitions