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 cₜ(N) be the number of t-core partitions of N. In 1999, Stanton conjectured cₜ(N) ≤ cₜ₊₁(N) if 4 ≤ t ≠ 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 cₜ(N), valid in all ranges of t and N. This includes the known asymptotic formulas for cₜ(N) as special cases, and shows that the behavior of cₜ(N) depends on how t² compares in size to N. For example, our formula implies that if t² = κ N + o(t), then cₜ(N) = exp(2π√A N)B N (1 + o(1)) for suitable constants A and B defined in terms of κ.