Permanental Inequalities for Totally Positive Matrices

Abstract

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 (2003), 442-470] concerning ratios of matrix minors. We also extend work of Drake, Gerrish, and the first author [Electron. J. Combin. 11 (2004), #N6] by characterizing the differences of monomials in $\mathbb{Z}[x_{1,1},x_{1,2},\dotsc,x_{n,n}]$ which evaluate positively on the set of all totally positive $n \times n$ matrices.