A Chinese Remainder Theorem for Partitions

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

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