A combinatorial proof of a relationship between maximal $(2k-1,2k+1)$ and $(2k-1,2k,2k+1)$-cores

TL;DR

This paper provides a combinatorial proof of a surprising numerical relationship between the sizes of maximal cores for specific integer partitions, connecting two types of core partitions through abacus theory and elementary number theory.

Contribution

It offers the first combinatorial interpretation of the size relationship between maximal $(2k-1,2k+1)$-cores and $(2k-1,2k,2k+1)$-cores, previously observed but not explained.

Findings

Established the size relationship: |κ_{2k-1,2k+1}| = 4|κ_{2k-1,2k,2k+1}|

Provided a combinatorial proof using abaci and partition dissection methods

Connected the result to properties of triangular numbers and squares

Abstract

Integer partitions which are simultaneously $t$--cores for distinct values of $t$ have attracted significant interest in recent years. When $s$ and $t$ are relatively prime, Olsson and Stanton have determined the size of the maximal $(s,t)$-core $\kappa_{s,t}$. When $k\geq 2$, a conjecture of Amdeberhan on the maximal $(2k-1,2k,2k+1)$-core $\kappa_{2k-1,2k,2k+1}$ has also recently been verified by numerous authors. In this work, we analyze the relationship between maximal $(2k-1,2k+1)$-cores and maximal $(2k-1,2k,2k+1)$-cores. In previous work, the first author noted that, for all $k\geq 1,$ $$ \vert \, \kappa_{2k-1,2k+1}\, \vert = 4\vert \, \kappa_{2k-1,2k,2k+1}\, \vert $$ and requested a combinatorial interpretation of this unexpected identity. Here, using the theory of abaci, partition dissection, and elementary results relating triangular numbers and squares, we provide such a combinatorial proof.