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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 Zx₁, ₁, x₁, ₂,. . . , x₍, ₍ which evaluate positively on the set of all totally positive n n matrices.
Skandera et al. (Sun,) studied this question.
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