论文标题
$ t $ cores和广场总和的组合结果
Combinatorial results on $t$-cores and sums of squares
论文作者
论文摘要
我们将连接与$ t $ cores和自轭$ t $ cores之间的连接与广场总和。为此，我们在$ t $ core分区和自换$ t $ t $ core分区之间提供了一个正整数$ n $的明确地图，以作为某些数字表示作为正方形的总和。例如，自轭$ 4 $ -CORE分区$λ=（4,1,1,1）$唯一对应于解决方案$ 61 = 6^2+5^2 $。作为推论，我们完全将$ t $ cores和hurwitz班级数量之间的关系分类。 使用这些工具，我们看到某些表示形式如何自然地将其分解成$ t $ cors的家庭。最后，我们在分区上构造了一个明确的地图，以解释平等$ 2 \ operatatorName {sc} _7（8n+1）= \ operatatorName {c} _4（7n+2）$先前由Bringmann，Kane和kane和第一作者研究。
We classify the connection between $t$-cores and self-conjugate $t$-cores to sums of squares. To do so, we provide explicit maps between $t$-core partitions and self-conjugate $t$-core partitions of a positive integer $n$ to representations of certain numbers as sums of squares. For example, the self-conjugate $4$-core partition $λ=(4,1,1,1)$ corresponds uniquely to the solution $61=6^2+5^2$. As a corollary, we completely classify the relationship between $t$-cores and Hurwitz class numbers. Using these tools, we see how certain sets of representations as sums of squares naturally decompose into families of $t$-cores. Finally, we construct an explicit map on partitions to explain the equality $2\operatorname{sc}_7(8n+1) = \operatorname{c}_4(7n+2)$ previously studied by Bringmann, Kane, and the first author.
