Sequentially congruent partitions and partitions into squares

In recent work, M. Schneider and the first author studied a curious class of integer partitions called “sequentially congruent” partitions: the $m$th 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 $p_{\mathrm{\square }}(n)$ 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 $k$th powers.