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Integer partitions which are simultaneously $t$--cores for distinct values of$t$ have attracted significant interest in recent years. When $s$ and $t$ arerelatively 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 themaximal $(2k-1,2k,2k+1)$-core $\kappa_{2k-1,2k,2k+1}$ has also recently beenverified by numerous authors.  In this work, we analyze the relationship between maximal $(2k-1,2k+1)$-coresand maximal $(2k-1,2k,2k+1)$-cores. In previous work, the first author notedthat, 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 interpretationof this unexpected identity. Here, using the theory of abaci, partitiondissection, and elementary results relating triangular numbers and squares, weprovide such a combinatorial proof.

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