Sequentially congruent partitions and partitions into squares

In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentiallyc congruent" partitions: the mth part is congruent to the (m + 1)th part modulo m, with the smallest part congruent to zero modulo the number of parts. Let p(S)(n) be the number of sequentially congruent partitions of n, and let p(square)(n) be the number of partitions of n wherein all parts are squares. In this note we prove bijectively, for all n >= 1, that p(S)(n) = p(square)(n). Our proof naturally extends to show other exotic classes of partitions of n are in bijection with certain partitions of n into kth powers.