On $t$-core and self-conjugate $(2t-1)$-core partitions in arithmetic progressions

Abstract

We extend recent results of Ono and Raji, relating the number of self-conjugate $7$-core partitions to Hurwitz class numbers. Furthermore, we give a combinatorial explanation for the curious equality $2{\mathrm{sc}}_7(8n+1)=c_4(7n+2)$. We also conjecture that an equality of this shape holds if and only if $t=4$, proving the cases $t\in \{2,3,5\}$ and giving partial results for $t>5$.