On core and bar-core partitions

TL;DR

This paper generalizes Olsson's 2008 results by removing the coprimality restriction, showing that the core and bar-core operations preserve their properties even when $s$ and $t$ are not relatively prime.

Contribution

It extends the theory of core and bar-core partitions by proving that the preservation properties hold without the coprimality condition.

Findings

Core and bar-core partitions are preserved under certain operations without coprimality.

Generalized results apply to broader classes of partitions.

Theoretical proof of the extended properties.

Abstract

If $s$ and $t$ are relatively prime J. Olsson proved in 2008 that the $s$-core of a $t$-core partition is again a $t$-core partition, and that the $s$-bar-core of a $t$-bar-core partition is again a $t$-bar-core partition. Here generalized results are proved for partitions and bar-partitions when the restriction that $s$ and $t$ be relatively prime is removed.