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

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)-coreκ_s,t. When k ≥ 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 ≥ 1, |κ_2k−1,2k+1 | = 4|κ_{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.