Monomial nonnegativity and the Bruhat order

We show that five nonnegativity properties of polynomials coincide when restricted to polynomials of the form x(1,pi(1)) ... x(n,pi(n)) - x(1,sigma(1)) ... x(n,sigma(n)), where pi and sigma are permutation s in S-n. In particular, we show that e a c h of these properties may be used to characterize the Bruhat order on S-n.