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

Rishi Nath¹ ; James A. Sellers²

¹Department of Mathematics, York College, City University of New York
²Department of Mathematics, Penn State University

The electronic journal of combinatorics, Tome 23 (2016) no. 1

Résumé

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 $k\geq 2$, 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\geq 1,$$$\vert \, \kappa_{2k-1,2k+1}\, \vert = 4\vert \, \kappa_{2k-1,2k,2k+1}\, \vert$$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.

Détail

DOI : 10.37236/5333

Classification : 05A17, 05E10, 20B30
Mots-clés : Young diagrams, symmetric group, \(p\)-cores, abaci, triangular numbers