A combinatorial proof of a relationship between maximal $(2k-1,2k+1)$ and $(2k-1,2k,2k+1)$-cores
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| Format: | Preprint |
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2015
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| author | Nath, Rishi Sellers, James A. |
| author_facet | Nath, Rishi Sellers, James A. |
| contents | 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 $κ_{s,t}$. When $k\geq 2$, 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 $k\geq 1,$ $$ \vert \, κ_{2k-1,2k+1}\, \vert = 4\vert \, κ_{2k-1,2k,2k+1}\, \vert $$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1506_06186 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | A combinatorial proof of a relationship between maximal $(2k-1,2k+1)$ and $(2k-1,2k,2k+1)$-cores Nath, Rishi Sellers, James A. Combinatorics 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 $κ_{s,t}$. When $k\geq 2$, 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 $k\geq 1,$ $$ \vert \, κ_{2k-1,2k+1}\, \vert = 4\vert \, κ_{2k-1,2k,2k+1}\, \vert $$ 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. |
| title | A combinatorial proof of a relationship between maximal $(2k-1,2k+1)$ and $(2k-1,2k,2k+1)$-cores |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1506.06186 |