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 ${\kappa }_{s,t}$. When $k\ge 2$, 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 $k\ge 1,$ $$|{\kappa }_{2k-1,2k+1}|=4|{\kappa }_{2k-1,2k,2k+1}|$$ 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.

10 pages, 6+ figures