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 ) ≤ c t + 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 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 = κ N + o ( t ) , then c t ( N ) = exp ⁡ ( 2 π A N ) B N ( 1 + o ( 1 ) ) for suitable constants A and B defined in terms of κ .