Combinatorial results on $t$-cores and sums of squares

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 $\lambda=(4,1,1,1)$ 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.