  In this paper we introduce discrete polymatroids satisfying the one-sided
strong exchange property and show that they are sortable (as a consequence
their base rings are Koszul) and that they satisfy White's conjecture. Since
any pruned lattice path polymatroid satisfies the one-sided strong exchange
property, this result provides an alternative proof for one of the main
theorems of J. Schweig in \cite{Sc}, where it is shown that every pruned
lattice path polymatroid satisfies White's conjecture. In addition, for two
classes of pruned lattice path polymatroidal ideals $I$ and their powers we
determine their depth and their associated prime ideals, and furthermore
determine the least power $k$ for which $\depth S/I^k$ and $\Ass(S/I^k)$
stabilize. It turns out that $\depth S/I^k$ stabilizes precisely when if
$\Ass(S/I^k)$ stabilizes.
