Sequentially congruent partitions and related bijections.

We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the length of the partition. It turns out these obscure-seeming objects are embedded in a natural way in partition theory. We show that sequentially congruent partitions are in bijective correspondence with the set of all partitions; moreover, sequentially congruent partitions with largest part $n$ are in bijection with the partitions of $n$. Finally, we show sequentially congruent partitions induce a bijection between partitions of $n$ and partitions of length $n$ whose parts obey a strict "frequency congruence" condition --- the frequency (or multiplicity) of each part is divisible by the part itself --- and prove infinite families of similar bijections, connecting with G. E. Andrews's theory of partition ideals.