A hyper m-ary partition of an integer n is defined to be a partition of n where each part is a power of m and each distinct power of m occurs at most m times. Let h_m(n) denote the number of hyper m-ary partitions of n and consider the resulting sequence. We show that the hyper m₁-ary partition sequence is a subsequence of the hyper m₂-ary partition sequence, for 2 ≤ m₁ ≤ m₂.