Asymptotics for $t$-Core Partitions and Stanton's Conjecture
Matthew Tyler
TL;DR
The paper resolves Stanton's conjecture by deriving a universal saddle-point asymptotic for the t-core partition count c_t(N) that remains valid across all t and N. Central to the approach is expressing c_t(N) as a contour integral of f_t(z) = η(tz)^t/η(z) and selecting a saddle parameter y via (μ_1(ity) − μ_1(iy))/y^2 = N + (t^2−1)/24, which yields a main Gaussian term with explicit prefactors depending on μ_2 and f_t(iy). The authors treat all regimes, including small, middle, and large t, deriving explicit formulas and error terms; the middle-t corollary connects to Hardy–Ramanujan-type growth with parameters A(κ) and B(κ). They also provide precise difference asymptotics c_{t+1}(N+t) − c_t(N+t) and use these to establish Stanton's inequality strictly for 4 ≤ t < N−1, effectively proving the conjecture in full. The work unifies prior fixed-t and large-N results and extends the understanding of t-core partitions across the entire parameter landscape, with rigorous explicit bounds enabling computational verification when needed.
Abstract
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 $t^2 = κN + o(t)$, then $c_t(N) = \frac{\exp\left(2π\sqrt{A N}\right)}{B N} (1 + o(1))$ for suitable constants $A$ and $B$ defined in terms of $κ$.