Discrete Polymatroids satisfying a stronger symmetric exchange property

TL;DR

This paper introduces discrete polymatroids with a stronger exchange property, proving they are sortable, Koszul, and satisfy White's conjecture, with applications to lattice path polymatroids and their ideals.

Contribution

It establishes that discrete polymatroids with a one-sided strong exchange property are sortable and satisfy White's conjecture, providing new proofs and insights into lattice path polymatroids.

Findings

Discrete polymatroids with the property are sortable and Koszul.

Pruned lattice path polymatroids satisfy White's conjecture.

Determined the depth and associated primes of certain polymatroidal ideals and their powers.

Abstract

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.