A Combinatorial Proof of a Relationship Between Maximal $(2k-1,2k+1)$-Cores and $(2k-1,2k,2k+1)$-Cores

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.