On the polymatroidal property of monomial ideals with a view towards orderings of minimal generators

We prove that a monomial ideal $I$ generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also conjecture that the polymatroidal ideals can be characterized with linear quotients property with respect to the reverse lexicographical ordering of the minimal generators induced by every ordering of variables. We prove our conjecture in many special cases.