On the existence of rook equivalent t-cores

摘要

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 divisible by t. In 1998, Haglund et al. (J. Combin. Theory Ser. A 84 (1) (1998) 9) proved that if t = 2, 3, or 4, then two distinct t-cores are rook equivalent if and only if they are conjugates. In contrast to this theorem, they conjectured that if t greater than or equal to 5, then there exists a constant N(t) such that for every positive integer n greater than or equal to N(t), there exist two distinct rook equivalent t-cores of n which are not Conjugate. Here this conjecture is proven for t greater than or equal to 12 with N(t) = 4 in all cases. (C) 2004 Elsevier Inc. All rights reserved.