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

TL;DR

This paper investigates a special symmetry in maximal $(s-1,s+1)$-core partitions, revealing structural properties using the $s$-abacus and extending the symmetry to a broader family of partitions.

Contribution

It explains the symmetry observed in maximal $(s-1,s+1)$-cores and generalizes these properties to a wider class of partitions using abacus techniques.

Findings

Maximal $(s-1,s+1)$-cores have empty $s$-core and $s$-quotients of 2-cores.

The symmetry imposes strong structural conditions on these partitions.

A broader family of partitions exhibiting similar symmetries is identified.

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.