An asymptotic formula for the t-core partition function and a conjecture of Stanton

For a positive integer t, a partition is said to be a t-core if each of the hook numbers from its Ferrers-Young diagram is not a multiple of t. In 1996, Granville and Ono proved the t-core partition conjecture, that a(t) (n), the number of t-core partitions of n, is positive for every nonnegative integer it as long as t >= 4. As part of their proof, they showed that if p >= 5 is prime, the generating function for a(p)(n) is essentially a multiple of an explicit Eisenstein Series together with a cusp form. This representation of the generating function leads to an asymptotic formula for a(p)(n) involving L-functions and divisor functions. In 1999, Stanton conjectured that for t >= 4 and n >= t + 1, a(t) (n) <= a(t+1) (n). Here we prove a weaker form of this conjecture, that for t >= 4 and it sufficiently large, a(t)(n) <= a(t+1)(n). Along the way, we obtain an asymptotic formula for at(n) which, in the cases where t is coprime to 6, is a generalization of the formula which follows from the work of Granville and Ono when t = p >= 5 is prime. (C) 2007 Elsevier Inc. All rights reserved.