A Chinese Remainder Theorem for Partitions

Let $s,t$ be natural numbers, and fix an $s$-core partition $\sigma$ and a $t$-core partition $\tau$. Put $d=\gcd(s,t)$ and $m= lcm(s,t)$, and write $N_{\sigma, \tau}(k)$ for the number of $m$-core partitions of length no greater than $k$ whose $s$-core is $\sigma$ and $t$-core is $\tau$. We prove that for $k$ large, $N_{\sigma, \tau}(k)$ is a quasipolynomial of period $m$ and degree $\frac{1}{d}(s-d)(t-d)$.