  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 with largest part $n$ are in
bijection with the partitions of $n$. Moreover, 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 that part -- and prove
families of similar bijections, connecting with G. E. Andrews's theory of
partition ideals.
