A Combinatorial Proof of a Relationship Between Maximal (2k-1,2k+1)-cores and (2k-1,2k,2k+1)-cores
Speaker: James Sellers, Penn State
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. When k > 1, a conjecture of Amdeberhan on the maximal (2k-1,2k,2k+1)-core 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, Nath noted that, for all k > 0, the size of the maximal (2k-1,2k+1)-core is exactly four times the size of the maximal (2k-1,2k,2k+1)-core 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. This is joint work with Rishi Nath.
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