Discrete polymatroids satisfying a stronger symmetric exchange property

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 [12], where it is shown that every pruned lattice path polymatroid satisfies White's conjecture. In addition we characterize a class of such polymatroids whose base rings are Gorenstein. Finally 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 depthS/Ik and Ass(S/Ik) stabilize. It turns out that depthS/Ik stabilizes precisely when Ass(S/Ik) stabilizes in both cases.