Symmetry in maximal $(s-1,s+1)$ cores

We explain a "curious symmetry" for maximal $(s-1,s+1)$-core partitions first observed by T. Amdeberhan and E. Leven. Specifically, using the $s$-abacus, we show such partitions have empty $s$-core and that their $s$-quotient is comprised of 2-cores. This imposes strong conditions on the partition structure, and implies both the Amdeberhan-Leven result and additional symmetry. We also find a more general family that exhibits these symmetries.

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