Combinatorial results on t-cores and sums of squares
Abstract
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=62+52. 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 2sc7(8n+1) = c4(7n+2) previously studied by Bringmann, Kane, and the first author.