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

Combinatorics/Partitions Seminar

Meeting Details

For more information about this meeting, contact James Sellers, Matthew Katz, Stephanie Geyer, George Andrews.

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.

Room Reservation Information

Room Number: 106 McAllister

Date: 09/01/2015

Time: 11:15am - 12:05pm