For 1 l j, let _l = _h=1^q(l){a_lh + Mv: v = 0, 1, 2,…}, where j, M, q(l) and the a_lh are positive integers such that j > 1, a_l1 < … < a_lq(2) M, and let _l^′ = _l {0}. Let p(n : ) be the number of partitions of n = (n₁,…,n_j) where, for 1 l j, the lth component of each part belongs to _l and let p^*(n : ) be the number of partitions of n into different parts where again the lth component of each part belongs to _l. Asymptotic formulas are obtained for p(n : ), p^*(n : ) where all but one n_l tend to infinity much more rapidly than that n_l, and asymptotic formulas are also obtained for p(n : ′), p^*(n ; ′), where one n_l tends to infinity much more rapidly than every other n_l. These formulas contrast with those of a recent paper (Robertson and Spencer, Trans. Amer. Math. Soc., to appear) in which all the n_l tend to infinity at approximately the same rate.