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

Abstract

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.

References in corpus (4)

• Armstrong's Conjecture for $(k,mk+1)$-Core Partitions
• Containment in (s,t)-core Partitions
• On the largest size of $(t,t+1,...,t+p)$-core partitions
• On the Enumeration of $(s,s+1,s+2)$-Core Partitions