A combinatorial proof of a relationship between maximal $(2k-1,2k+1)$ 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 $κ_{s,t}$. When $k\geq 2$, a conjecture of Amdeberhan on the maximal $(2k-1,2k,2k+1)$-core $κ_{2k-1,2k,2k+1}$ 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 $k\geq 1,$ $$ \vert \, κ_{2k-1,2k+1}\, \vert = 4\vert \, κ_{2k-1,2k,2k+1}\, \vert $$ 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.

Figures (6)

• Figure 1: Young diagram of $(8,6,5^2,3,2^3,1)$
• Figure 2: 8-quotient of $\kappa_{7,9}$
• Figure 3: The $8$-abacus $\bar{\alpha}(8)$ of $\kappa_{7,8,9}$
• Figure 4: The $10$-abacus $\bar{\alpha}(10)$ of $\kappa_{9,10,11}$
• Figure 5: $T_5 + T_4 = 5^2$

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