Asymptotics for $t$-Core Partitions and Stanton's Conjecture

A partition is a $t$-core partition if $t$ is not one of its hook lengths. Let $c_t(N)$ be the number of $t$-core partitions of $N$. In 1999, Stanton conjectured $c_t(N) \le c_{t+1}(N)$ if $4 \le t \ne N-1$. This was proved for $t$ fixed and $N$ sufficiently large by Anderson, and for small values of $t$ by Kim and Rouse. In this paper, we prove Stanton's conjecture in general. Our approach is to find a saddle point asymptotic formula for $c_t(N)$, valid in all ranges of $t$ and $N$. This includes the known asymptotic formulas for $c_t(N)$ as special cases, and shows that the behavior of $c_t(N)$ depends on how $t^2$ compares in size to $N$. For example, our formula implies that if $t^2 = \kappa N + o(t)$, then $c_t(N) = \frac{\exp\left(2\pi\sqrt{A N}\right)}{B N} (1 + o(1))$ for suitable constants $A$ and $B$ defined in terms of $\kappa$.