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 $$p_{{\mathcal {S}}}(n)$$ 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 $$n\ge 1,$$ that $$p_{{\mathcal {S}}}(n) = p_{\square }(n).$$ Our proof naturally extends to show other exotic classes of partitions of n are in bijection with certain partitions of n into kth powers.