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arXiv · 2406.00963

Permanental inequalities for totally positive matrices

Mark Skandera·Daniel Soskin
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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 [{\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.

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Mark Skandera, Daniel Soskin. 2024-06-03. Permanental inequalities for totally positive matrices. https://arxiv.org/abs/2406.00963

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