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 [{ \em Adv.\ Appl.\ Math.} { \bf 30} no.\ 3, (2003) pp.\ 442--470] concerning ratios of matrix minors. We also extend work of Drake, Gerrish, and the first author [{ \em Electron.\ J.\ Combin.,} { \bf 11} no.\ 1, (2004) Note 6] by characterizing the differences of monomials in $\mathbb{ Z}[x_{ 1,1},x_{ 1,2},...,x_{ n,n}]$ which evaluate positively on the set of all totally positive $n \times n$ matrices.
Categories: math.CO