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

作者：Nath Rishi^*; Sellers James A^*

来源：Electronic Journal of Combinatorics, 2016, 23(1): P1.13.

DOI：10.37236/5333

• Young diagrams
• symmetric group
• p-cores
• abaci
• triangular numbers

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 K-s,K-t. When k >= 2, a conjecture of Amdeberhan on the maximal (2k - 1, 2k, 2k + 1)-core K-2k-1,K-2k,K-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,
vertical bar K-2k-1,K-2k+1 vertical bar = 4 vertical bar K-2k-1,K-2k,K-2k+1 vertical bar
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.

• 出版日期2016-1-22