  In this paper we study partitions whose successive ranks belong to a given
set. We enumerate such partitions while keeping track of the number of parts,
the largest part, the side of the Durfee square, and the height of the Durfee
rectangle. We also obtain a new bijective proof of a result of Andrews and
Bressoud that the number of partitions of $N$ with all ranks at least $1-\ell$
equals the number of partitions of $N$ with no parts equal to $\ell+1$, for
$\ell\ge0$, which allows us to refine it by the above statistics. Combining
Foata's second fundamental transformation for words with Greene and Kleitman's
mapping for subsets, interpreted in terms of lattice paths, we obtain
enumeration formulas for partitions whose successive ranks satisfy certain
constraints, such as being bounded by a constant.
