  We characterize the downsets of integer partitions (ordered by containment of
Ferrers diagrams) and compositions (ordered by the generalized subword order)
which have finite dimension in the sense of Dushnik and Miller. In the case of
partitions, while the set of all partitions has infinite dimension, we show
that every proper downset of partitions has finite dimension. For compositions
we identify four minimal downsets of infinite dimension and establish that
every downset which does not contain one of these four has finite dimension.
