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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 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.fatcat:ulb5r6xyxjdmbeo437i5uvd6jq