Permanental Inequalities for Totally Positive Matrices

We characterize ratios of permanents of (generalized) submatrices which are bounded on the set of all totally positive matrices. This provides a permanental analog of results of Fallat, Gekhtman, and Johnson [Adv. Appl. Math. 30 no. 3, (2003) pp. 442-470] concerning ratios of matrix minors. We also extend work of Drake, Gerrish, and the first author [Electron. J. Combin., 11 no. 1, (2004) Note 6] by characterizing the differences of monomials in Z[x1,1, x1,2, ... , xn,n] which evaluate positively on the set of all totally positive n x n matrices.