%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2025}
\articlenumber{124}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}

\usepackage{lipsum}

%% define your title in the usual way
\title
%[Pemanental inequalities]
{Multiplicative permanental inequalities for totally positive matrices}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author{\texorpdfstring{Mark Skandera%\thanks{\href{mailto:hello@world.c}{hello@world.c}.}
\addressmark{1} \and Daniel Soskin\addressmark{1}}{Mark Skandera%
 and Daniel Soskin}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Lehigh University
%\\ \addressmark{2} Department of Mathematics, Lehigh University
}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet

%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{We characterize ratios of permanents of submatrices which are bounded 
%as real-valued functions 
on the set of totally positive matrices.  This provides a permanental
analog of results of Fallat, Gekhtman, and Johnson
%et.\ al. 
[{\em Adv.\ Appl.\ Math.}, {\bf 30} no.\ 3, (2003)]
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)] by characterizing differences of monomials in $\zxn$ which evaluate positively on all
%the set of 
totally positive $n \times n$ matrices.}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{total positivity, permanent, inequalities}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex, giveninits]{biblatex}
\appto\bibfont{\setlength{\emergencystretch}{.5em}}

\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{obs}[thm]{Observation}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{prob}[thm]{Problem}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}%[section]
\newtheorem{alg}[thm]{Algorithm}

\newcommand{\sn}{\mathfrak{S}_n}
\newcommand{\sr}{\mathfrak{S}_r}
\newcommand{\mfs}[1]{\mathfrak{S}_{#1}}
\newcommand{\snn}{\mfs{\smash{[\ol n, n]}}}
\newcommand{\mfsmin}[1]{\mathfrak{S}_-^{#1}}
\newcommand{\bn}{\mathfrak{B}_n}
\newcommand{\br}{\mathfrak{B}_r}
\newcommand{\mfb}[1]{\mathfrak{B}_{#1}}
\newcommand{\mfbmin}[1]{\mathfrak{B}_-^{#1}}
\newcommand{\zsn}{\mathbb{Z}[\sn]}
\newcommand{\qsn}{\mathbb{Q}[\sn]}
\newcommand{\zbn}{\mathbb{Z}[\bn]}
\newcommand{\qbn}{\mathbb{Q}[\bn]}
\newcommand{\cx}{\mathbb{C}[x]}
\newcommand{\zx}{\mathbb{Z}[x]}
\newcommand{\cxn}{\mathbb{C}[x_{1,1},x_{1,2},\dotsc,x_{n,n}]}
\newcommand{\zxn}{\mathbb{Z}[x_{1,1},x_{1,2},\dotsc,x_{n,n}]}
\newcommand{\cqq}{\mathbb{C}[\qp12, \qm12]}
\newcommand{\zqq}{\mathbb{Z}[\qp12, \qm12]}
\newcommand{\zqQ}{\mathbb{Z}[\qp12, \qm12, \Qp12, \Qm12]}
\newcommand{\nqq}{\mathbb{N}[\qp12, \qm12]}
\newcommand{\czn}{\cqq \langle z_{1,1},\dotsc,z_{n,n} \rangle}
%\newcommand{\cz}{\cqq \langle z \rangle}
\newcommand{\cz}{\mathbb{C}[z]}
\newcommand{\zz}{\mathbb{Z}[z]}
\newcommand{\cyr}{\mathbb{C}[y_{1,1},\dotsc,x_{r,r}]}
\newcommand{\csn}{\mathbb{C}[\sn]}
\newcommand{\bitab}{\mathcal B}
\newcommand{\tab}{\mathcal U}
%\newcommand{\bctab}{\mathcal U^*}
\newcommand{\bctab}{\mathcal U^{\msfBC}}
\newcommand{\atab}{\mathcal U^{\msfA}}
\newcommand{\tp}[1]{\mathcal M^{\textsc{TP}}_{#1}}
\newcommand{\tnn}[1]{\mathcal M^{\textsc{TNN}}_{#1}}
\newcommand{\hps}[1]{\mathcal M^{\textsc{HPS}}_{#1}}

\newcommand{\Ie}{{I, \emptyset}}
\newcommand{\eJ}{{\emptyset, J}}
\newcommand{\Je}{{J, \emptyset}}
\newcommand{\eI}{{\emptyset, I}}
\newcommand{\IIJJ}{{I, J}}

\newcommand{\wo}{w_0}
\newcommand{\woi}{w_0^I}
\newcommand{\woj}{w_0^J}
\newcommand{\wok}{w_0^K}
\newcommand{\wokp}{w_0^{K'}}

\newcommand{\etu}{\epsilon_{t,u}}
\newcommand{\etv}{\epsilon_{t,v}}
\newcommand{\etz}{\epsilon_{t,z}}
\newcommand{\eut}{\epsilon_{u,t}}
\newcommand{\euv}{\epsilon_{u,v}}
\newcommand{\euw}{\epsilon_{u,w}}
\newcommand{\euy}{\epsilon_{u,y}}
\newcommand{\euz}{\epsilon_{u,z}}
\newcommand{\evu}{\epsilon_{v,u}}
\newcommand{\evw}{\epsilon_{v,w}}
\newcommand{\evy}{\epsilon_{v,y}}
\newcommand{\evz}{\epsilon_{v,z}}
\newcommand{\ewy}{\epsilon_{w,y}}
\newcommand{\ewz}{\epsilon_{w,z}}
\newcommand{\eyz}{\epsilon_{y,z}}
\newcommand{\ezu}{\epsilon_{z,u}}
\newcommand{\ezw}{\epsilon_{z,w}}
\newcommand{\ezy}{\epsilon_{z,y}}
\newcommand{\eeu}{\epsilon_{e,u}}
\newcommand{\eev}{\epsilon_{e,v}}
\newcommand{\eew}{\epsilon_{e,w}}

\newcommand{\qp}[2]{q^{\frac{#1}{#2}}}
\newcommand{\qm}[2]{q^{\negthinspace\Bar\,\frac{#1}{#2}}}
\newcommand{\Qp}[2]{Q^{\frac{#1}{#2}}}
\newcommand{\Qm}[2]{Q^{\negthinspace\Bar\,\frac{#1}{#2}}}
\newcommand{\qtu}{q_{t,u}}
\newcommand{\qtv}{q_{t,v}}
\newcommand{\qtz}{q_{t,z}}
\newcommand{\quv}{q_{u,v}}
\newcommand{\quw}{q_{u,w}}
\newcommand{\quy}{q_{u,y}}
\newcommand{\qvu}{q_{v,u}}
\newcommand{\qvw}{q_{v,w}}
\newcommand{\qvy}{q_{v,y}}
\newcommand{\qvz}{q_{v,z}}
\newcommand{\qyw}{q_{y,w}}
\newcommand{\qwy}{q_{w,y}}
\newcommand{\qwz}{q_{w,z}}
\newcommand{\qzw}{q_{z,w}}
%\newcommand{\qeu}{q_{e,u}}
\newcommand{\qeu}{q_u}
\newcommand{\qet}{q_t}
\newcommand{\qey}{q_y}
%\newcommand{\qev}{q_{e,v}}
\newcommand{\qev}{q_v}
%\newcommand{\qew}{q_w}
\newcommand{\qew}{\smash{\qp{\ell(w)}2}}
\newcommand{\qitu}{q_{t,u}^{-1}}
\newcommand{\qitv}{q_{t,v}^{-1}}
\newcommand{\qitw}{q_{t,w}^{-1}}
\newcommand{\qity}{q_{t,y}^{-1}}
\newcommand{\qiut}{q_{u,t}^{-1}}
\newcommand{\qiuv}{q_{u,v}^{-1}}
\newcommand{\qiuy}{q_{u,y}^{-1}}
\newcommand{\qiuw}{q_{u,w}^{-1}}
\newcommand{\qiuz}{q_{u,z}^{-1}}
\newcommand{\qivu}{q_{v,u}^{-1}}
\newcommand{\qivw}{q_{v,w}^{-1}}
\newcommand{\qivy}{q_{v,y}^{-1}}
\newcommand{\qivz}{q_{v,z}^{-1}}
\newcommand{\qiyw}{q_{y,w}^{-1}}
\newcommand{\qiyz}{q_{y,z}^{-1}}
\newcommand{\qizw}{q_{z,w}^{-1}}
\newcommand{\qizy}{q_{z,y}^{-1}}
%\newcommand{\qiet}{q_{e,t}^{-1}}
%\newcommand{\qieu}{q_{e,u}^{-1}}
%\newcommand{\qiev}{q_{e,v}^{-1}}
%\newcommand{\qiew}{q_{e,w}^{-1}}
\newcommand{\qiew}{\qm{\ell(w)}2}
%\newcommand{\qiew}{q_w^{-1}}
\newcommand{\qiey}{q_y^{-1}}
\newcommand{\qiev}{q_v^{-1}}
%\newcommand{\qiey}{q_{e,y}^{-1}}
\newcommand{\qieyp}{q_{y'}^{-1}}
\newcommand{\qiez}{q_{e,z}^{-1}}
\newcommand{\qdiff}{\qp12 - \qm12}

\newcommand{\wT}{\widetilde T}
\newcommand{\wTp}{\widetilde T'}
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\wR}[1]{\widetilde R_{#1}}
\newcommand{\wS}[1]{\widetilde S_{#1}}
\newcommand{\wSp}{\widetilde S'}

\newcommand{\hnq}{H_n(q)}
%\newcommand{\hnq}{H_n^{\sfA}(q)}
\newcommand{\hbnq}{H_n^{\msfBC}(q)}
%\newcommand{\hbnq}{H_n^{\mathfrak{B}}(q)}
\newcommand{\hbq}[1]{H_{#1}^{\msfBC}(q)}
\newcommand{\hbnQ}{H_n^C(q,Q)}
\newcommand{\hrq}{H_r(q)}
\newcommand{\hlq}{H_\lambda(q)}
%\newcommand{\anq}{\mathcal{A}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\anq}{\mathcal{A}(n,q)}
\newcommand{\atwonq}{\mathcal{A}(2n,q)}
\newcommand{\arq}{\mathcal{A}(r;q)}
%\newcommand{\arnq}{\mathcal{A}_r(n;q)}
\newcommand{\arnq}{\mathcal{A}_r}
\newcommand{\annq}{\mathcal{A}_n(n;q)}
%\newcommand{\annnq}{\mathcal{A}_{[n],[n]}(n;q)}
\newcommand{\Ann}{\mathcal{A}_{[n],[n]}}
\newcommand{\annnq}{\mathcal{A}_{[n],[n]}}
\newcommand{\anmnq}{\mathcal{A}_{[n],M}(n;q)}
\newcommand{\amnnq}{\mathcal{A}_{M,[n]}(n;q)}
\newcommand{\alnnq}{\mathcal{A}_{L,[n]}(n;q)}
%\newcommand{\almnq}{\mathcal{A}_{L,M}(n;q)}
\newcommand{\AnM}{\mathcal{A}_{[n],M}}
\newcommand{\ALM}{\mathcal{A}_{L,M}}
\newcommand{\amlnq}{\mathcal{A}_{M,L}(n;q)}
\newcommand{\arrrq}{\mathcal{A}_{[r],[r]}(r;q)}
\newcommand{\alrrq}{\mathcal{A}_{L,[r]}(r;q)}
\newcommand{\armrq}{\mathcal{A}_{[r],M}(r;q)}
\newcommand{\trsp}{\mathcal{T}}
\newcommand{\btrsp}{\mathcal{T}^{BC}}

\newcommand{\coh}{\mathrm{H}}
\newcommand{\icoh}{\mathrm{IH}}
\newcommand{\HH}{\mathcal{H}}
\newcommand{\Hpie}{H'_{\smash{\Ie}}}
\newcommand{\Hie}{H_\Ie}
\newcommand{\Hpej}{H'_{\smash\eJ}}
\newcommand{\Hej}{H_\eJ}
\newcommand{\Hpje}{H'_\Je}
\newcommand{\Hje}{H_\Je}
\newcommand{\Hpei}{H'_\eI}
\newcommand{\Hei}{H_\eI}
\newcommand{\Hij}{H_{I,J}}
\newcommand{\Hpij}{H'_{I,J}}

\newcommand{\Wiep}{W^{I,\emptyset}_+}
\newcommand{\Wiem}{W^{I,\emptyset}_-}
\newcommand{\WIkpem}{(W_I)^{K',\emptyset}_-}
\newcommand{\WIekpm}{(W_I)^{\emptyset,K'}_-}
\newcommand{\WIekpp}{(W_I)^{\emptyset,K'}_+}
\newcommand{\WIenpm}{(W_I)^{\emptyset,N'}_-}
\newcommand{\WJiem}{(W_J)^{I,\emptyset}_-}
\newcommand{\WJkem}{(W_J)^{K,\emptyset}_-}
\newcommand{\WJkep}{(W_J)^{K,\emptyset}_+}
\newcommand{\WJnem}{(W_J)^{N,\emptyset}_-}
\newcommand{\Wjep}{W^{J,\emptyset}_+}
\newcommand{\Wjem}{W^{J,\emptyset}_-}
\newcommand{\Wkep}{W^{K,\emptyset}_+}
\newcommand{\Wkem}{W^{K,\emptyset}_-}
\newcommand{\Weip}{W^{\emptyset,I}_+}
\newcommand{\Weim}{W^{\emptyset,I}_-}
\newcommand{\WJeim}{(W_J)^{\emptyset,I}_-}
\newcommand{\WJekm}{(W_J)^{\emptyset,K}_-}
\newcommand{\WJenm}{(W_J)^{\emptyset,N}_-}
\newcommand{\Wejp}{W^{\emptyset,J}_+}
\newcommand{\Wejm}{W^{\emptyset,J}_-}
\newcommand{\Wekp}{W^{\emptyset,K}_+}
\newcommand{\Wekpp}{W^{\emptyset,K'}_+}
\newcommand{\Wekm}{W^{\emptyset,K}_-}
\newcommand{\Wekpm}{W^{\emptyset,K'}_-}
\newcommand{\Wijp}{W^{I,J}_+}
\newcommand{\Wijm}{W^{I,J}_-}

\newcommand{\Rie}[1]{R_{#1}^{I,\emptyset}}
\newcommand{\Rpie}[1]{R^{\prime\,I,\emptyset}_{#1}}
\newcommand{\Rej}[1]{R_{#1}^{\emptyset,J}}
\newcommand{\Rpej}[1]{R^{\prime\,\emptyset,J}_{#1}}
\newcommand{\Rij}[1]{R_{#1}^{I,J}}
\newcommand{\Rpij}[1]{R^{\prime\,I,J}_{#1}}
\newcommand{\wRie}[1]{\wR_{#1}^{I,\emptyset}}
\newcommand{\wRpie}[1]{\wR^{\prime\,I,\emptyset}_{#1}}
\newcommand{\wRej}[1]{\wR_{#1}^{\emptyset,J}}
\newcommand{\wRpej}[1]{\wR^{\prime\,\emptyset,J}_{#1}}
\newcommand{\wRij}[1]{\wR_{#1}^{I,J}}
\newcommand{\wRpij}[1]{\wR^{\prime\,I,J}_{#1}}
\newcommand{\Sie}[1]{S_{#1}^{I,\emptyset}}
\newcommand{\Spie}[1]{S^{\prime\,I,\emptyset}_{#1}}
\newcommand{\Sej}[1]{S_{#1}^{\emptyset,J}}
\newcommand{\Spej}[1]{S^{\prime\,\emptyset,J}_{#1}}
\newcommand{\Sij}[1]{S_{#1}^{I,J}}
\newcommand{\Spij}[1]{S^{\prime\,I,J}_{#1}}
\newcommand{\wSie}[1]{\wS_{#1}^{I,\emptyset}}
\newcommand{\wSpie}[1]{\wS^{\prime\,I,\emptyset}_{#1}}
\newcommand{\wSej}[1]{\wS_{#1}^{\emptyset,J}}
\newcommand{\wSpej}[1]{\wS^{\prime\,\emptyset,J}_{#1}}
\newcommand{\wSij}[1]{\wS_{#1}^{I,J}}
\newcommand{\wSpij}[1]{\wS^{\prime\,I,J}_{#1}}
\newcommand{\Qie}[1]{Q_{#1}^{I,\emptyset}}
\newcommand{\Qej}[1]{Q_{#1}^{\emptyset,J}}
\newcommand{\Qij}[1]{Q_{#1}^{I,J}}
\newcommand{\pie}[1]{p_{#1}^{I,\emptyset}}
\newcommand{\pje}[1]{p_{#1}^{J,\emptyset}}
\newcommand{\pei}[1]{p_{#1}^{\emptyset,I}}
\newcommand{\pej}[1]{p_{#1}^{\emptyset,J}}
\newcommand{\pij}[1]{p_{#1}^{I\ntnsp,J}}
\newcommand{\pji}[1]{p_{#1}^{J\ntnsp,I}}
\newcommand{\rie}[1]{r_{#1}^{I\ntnsp,\emptyset}}
\newcommand{\rje}[1]{r_{#1}^{J\ntnsp,\emptyset}}
\newcommand{\rei}[1]{r_{#1}^{\emptyset,I}}
\newcommand{\rej}[1]{r_{#1}^{\emptyset,J}}
\newcommand{\rij}[1]{r_{#1}^{I\ntnsp,J}}
\newcommand{\rji}[1]{r_{#1}^{J\ntnsp,I}}
\newcommand{\redexp}[2]{s_{i_{#1}} \ntnsp \cdots s_{i_{#2}}}
\newcommand{\wtc}[2]{\widetilde{C}_{#1}(#2)}
%\newcommand{\btc}[2]{C_{#1}^{\mathfrak B}(#2)}
\newcommand{\btc}[2]{\widetilde{C}_{#1}^{\msfBC}(#2)}

\newcommand{\imm}[1]{\mathrm{Imm}_{#1}}
\newcommand{\immlm}[1]{\mathrm{Imm}_{#1}^{L,M}}
\newcommand{\immln}[1]{\mathrm{Imm}_{#1}^{L,[n]}}
\newcommand{\immlr}[1]{\mathrm{Imm}_{#1}^{L,[r]}}
\newcommand{\immnm}[1]{\mathrm{Imm}_{#1}^{[n],M}}
\newcommand{\immrm}[1]{\mathrm{Imm}_{#1}^{[r],M}}
%\newcommand{\cimm}[1]{\text{$C$-}\mathrm{Imm}_{#1}}
\newcommand{\bnimm}[1]{\mathrm{Imm}_{#1}^{\bn}}
\newcommand{\simm}[2]{\mathrm{Imm}_{#2}^{\smash{\mfs{#1}}}}

\newcommand{\sumsb}[1]{\sum_{\substack{#1}}}  %%%for multi-line sums
\newcommand{\prodsb}[1]{\prod_{\substack{#1}}}  %%%for multi-line products
\newcommand{\inv}{\textsc{inv}}
\newcommand{\rinv}{\textsc{rinv}}
%\newcommand{\defeq}{\underset{\mathrm{def}}=}
\newcommand{\defeq}{:=}
%\newcommand{\dfct}{\textsc{dfct}}
\newcommand{\dfct}{\mathrm{dfct}}
\newcommand{\dnc}{\textsc{dnc}}
\newcommand{\dc}{\textsc{dc}}
\newcommand{\pnc}{\textsc{pnc}}
\newcommand{\sct}{\mathrm{sct}}
\newcommand{\des}{\mathrm{des}}
\newcommand{\exc}{\mathrm{exc}}
%\newcommand{\ct}{\mathrm{ctype}}
\newcommand{\stat}{\mathrm{stat}}
\newcommand{\frobch}{\mathrm{ch}}
\newcommand{\npfrobch}{\mathrm{nch}}
\newcommand{\pfrobch}{\mathrm{pch}}
\newcommand{\spn}{\mathrm{span}}
\newcommand{\sgn}{\mathrm{sgn}}
\newcommand{\triv}{\mathrm{triv}}
\newcommand{\wgt}{\mathrm{wgt}}
\newcommand{\src}{\mathrm{src}}
\newcommand{\snk}{\mathrm{snk}}
\newcommand{\type}{\mathrm{type}}
\newcommand{\ctype}{\mathrm{ctype}}
\newcommand{\inc}{\mathrm{inc}}
\newcommand{\ngr}{\mathrm{ngr}}
\newcommand{\incross}{\textsc{invnc}}
\newcommand{\cdncross}{\textsc{cdnc}}
\newcommand{\cross}{\textsc{c}}
\newcommand{\pavoiding}{$3412$-avoiding, $4231$-avoiding }
\newcommand{\avoidsp}{avoids the patterns $3412$ and $4231${}}
\newcommand{\avoidp}{avoid the patterns $3412$ and $4231${}}
\newcommand{\avoidingp}{avoiding the patterns $3412$ and $4231${}}
\newcommand{\theps}{the patterns $3412$ and $4231${}}
\newcommand{\avoidsignedp}{avoid the signed patterns $1\ol2$, $\ol21$,
$\ol2\ol1$, $312$, $3\ol12${}}
\newcommand{\avoidssignedp}{avoids the signed patterns $1\ol2$,
$\ol21$, $\ol2\ol1$, $312$, $3\ol12${}}
\newcommand{\avoidingsignedp}{avoiding the signed patterns $1\ol2$,
$\ol21$, $\ol2\ol1$, $312$, $3\ol12${}}
\newcommand{\ul}[1]{\underline{#1}}
\newcommand{\ssec}[1]{\subsection{#1}{$\negthinspace$}}

\newcommand{\tr}{{\negthickspace \top \negthickspace}}
\newcommand{\ntnsp}{\negthinspace}
\newcommand{\ntksp}{\negthickspace}
\newcommand{\nTksp}{\negthickspace\negthickspace}
\newcommand{\nTknsp}{\negthickspace\negthickspace\negthinspace}
\newcommand{\nTtksp}{\negthickspace\negthickspace\negthickspace}
\newcommand{\nTTksp}{\negthickspace\negthickspace\negthickspace
\negthickspace}
\newcommand{\nTTtksp}{\negthickspace\negthickspace\negthickspace
\negthickspace\negthickspace}
\newcommand{\bp}{\begin{prob}}
\newcommand{\ep}{\end{prob}}


\newcommand{\kbar}[2]{\varphi_#1(#2)}

\newcommand{\hnqq}{H_n(q_2,q_4)}
\newcommand{\anqq}{\mathcal{A}(n;q_2,q_4)}
\newcommand{\arnqq}{\mathcal{A}_r(n;q_2,q_4)}
\newcommand{\almnqq}{\mathcal{A}_{L,M}(n;q_2,q_4)}
\newcommand{\annnqq}{\mathcal{A}_{[n],[n]}(n;q_2,q_4)}
\newcommand{\aT}{\widetilde{\mathcal{T}}}
\newcommand{\what}[1]{\widehat{#1}}
\newcommand{\q}[3]{q_{#1,#2,#3}}
\newcommand{\qi}[3]{q^{-1}_{#1,#2,#3}}
\newcommand{\qsum}{q_2 + q_4}
\newcommand{\qprod}{-q_2q_4}
\newcommand{\qprodi}{(-q_2q_4)^{-1}}

%\newcommand{\oglnc}{{\mathcal O}(GL(n,\mathbb{C}))}
%\newcommand{\oqglnc}{{\mathcal O}_q(GL(n,\mathbb{C}))}
%\newcommand{\oslnc}{{\mathcal O}(SL(n,\mathbb{C}))}
%\newcommand{\oslthreec}{{\mathcal O}(SL(3,\mathbb{C}))}
%\newcommand{\oqslnc}{{\mathcal O}_q(SL(n,\mathbb{C}))}
\newcommand{\oglnc}{{\mathcal O}(\mathrm{GL}_n (\mathbb C))}
\newcommand{\oqglnc}{{\mathcal O}_q(\mathrm{GL}_n (\mathbb C))}
\newcommand{\oslnc}{{\mathcal O}(\mathrm{SL}_n) (\mathbb C)}
\newcommand{\oslthreec}{{\mathcal O}(\mathrm{SL}_3 (\mathbb C))}
\newcommand{\oqslnc}{{\mathcal O}_q(\mathrm{SL}_n (\mathbb C))}
\newcommand{\ospnc}{{\mathcal O}(\mathrm{SP}_{2n} (\mathbb C))}
\newcommand{\oqspnc}{{\mathcal O}_q(\mathrm{SP}_{2n} (\mathbb C))}
\newcommand{\uglnc}{U(\mathfrak{gl}(n,\mathbb{C}))}
\newcommand{\uqglnc}{U_q(\mathfrak{gl}(n,\mathbb{C}))}
\newcommand{\uslnc}{U(\mathfrak{sl}(n,\mathbb{C}))}
\newcommand{\uqslnc}{U_q(\mathfrak{sl}(n,\mathbb{C}))}
\newcommand{\uspnc}{U(\mathfrak{sp}_{2n}(\mathbb{C}))}
\newcommand{\uqspnc}{U_q(\mathfrak{sp}_{2n}(\mathbb{C}))}
\newcommand{\mat}[1]{\mathrm{Mat}_{#1 \times #1}}
\newcommand{\spc}[1]{\mathfrak{sp}_{#1}(\mathbb{C})}
\newcommand{\slc}[1]{\mathfrak{sl}_{#1}(\mathbb{C})}
\newcommand{\fl}[1]{\mathcal{F}_{#1}}
%\newcommand{\hess}[1]{\mathcal{H}({\mathbf #1})}
\newcommand{\PiBC}{\Pi^{\msfBC}}


\newcommand{\qdet}{\mathrm{det}_q}
\newcommand{\qper}{\mathrm{per}_q}
\newcommand{\perm}{\mathrm{per}}
\newcommand{\rnk}{\mathrm{rank}}
\newcommand{\sort}{\mathrm{sort}}
\newcommand{\shape}{\mathrm{shape}}
\newcommand{\hess}{\mathrm{Hess}}
\newcommand{\ahess}{\mathrm{Hess^{\msfA}}}
\newcommand{\bhess}{\mathrm{Hess}^{\msfBB}}
\newcommand{\chess}{\mathrm{Hess}^{\msfC}}
%\newcommand{\exp}{\mathrm{exp}}
\newcommand{\permmon}[2]{#1_{1,#2_1} \ntnsp\cdots #1_{n,#2_n}}
\newcommand{\bpermmon}[2]
           {#1_{\smash{\ol n,#2_{\ol n}}} \ntnsp\cdots #1_{\smash{\ol
1,#2_{\ol 1}}}
           #1_{1,#2_{1}} \ntnsp\cdots #1_{n,#2_n} }
\newcommand{\sprod}[2]{s_{#1_1} \ntnsp\cdots s_{#1_#2}}
\newcommand{\sintprod}[3]{s_{[#1_1,#2_1]} \ntnsp\cdots s_{[#1_#3, #2_#3]}}
\newcommand{\ssm}{\smallsetminus}
\newcommand{\multiu}{\Cup}
\newcommand{\bs}{\backslash}
\newcommand{\bfx}{\mathbf x}
\newcommand{\wleq}{\leq_W}
\newcommand{\wless}{<_W}
\newcommand{\slambda}{\mathfrak{S}_\lambda}
\newcommand{\slambdamin}{\mathfrak{S}_\lambda^{-}}
\newcommand{\phmin}{\phantom{-}}
%\newcommand{\circd}[1]{\raisebox{.5pt}{\textcircled{\raisebox{-.9pt}{#1}}}}
\newcommand{\circd}[1]{\raisebox{-9pt}{\textcircled{\raisebox{-.9pt}{#1}}}}
%\newcommand{\oglnc}{{\mathcal O}(GL(n,\mathbb{C}))}
%\newcommand{\oqglnc}{{\mathcal O}_q(GL(n,\mathbb{C}))}
%\newcommand{\oslnc}{{\mathcal O}(SL(n,\mathbb{C}))}
%\newcommand{\oqslnc}{{\mathcal O}_q(SL(n,\mathbb{C}))}
\newcommand{\sdot}{\left[ \begin{smallmatrix} 0 & -1 \\ 1 &
\phantom{-}0 \end{smallmatrix}\right]}
\newcommand{\sddot}{\left[ \begin{smallmatrix} \phantom{-}0 & 1 \\ -1
& 0 \end{smallmatrix}\right]}
\newcommand{\sreg}{\left[ \begin{smallmatrix} 0 & 1 \\ 1 & 0
\end{smallmatrix}\right]}
%\newcommand{\smallj}{\left[ \begin{smallmatrix} 0 & & 1 \\  &
%\text{\raisebox{1mm}{$\iddots$}} & \\ 1 & & 0
%\end{smallmatrix}\right]}
\newcommand{\smallj}{\left[ \begin{smallmatrix} 0 & \phantom{\int} & 1
\\ & \smash{\iddots} & \phantom{T} \\ 1 & & 0
\end{smallmatrix}\right]}
\newcommand{\abdiag}[1]{\left[ \begin{smallmatrix} 1 & #1 \\ 0 & 1
\end{smallmatrix}\right]}
\newcommand{\bediag}[1]{\left[ \begin{smallmatrix} 1 & 0 \\ #1 & 1
\end{smallmatrix}\right]}
\newcommand{\upparrow}{\big \uparrow \nTksp \phantom{\uparrow}}
\newcommand{\upparrowa}{\big \uparrow \ntksp \phantom{\uparrow}}
\newcommand{\precdot}{\prec \negthickspace \negthinspace \cdot\ }
\newcommand{\net}[2]{\mathcal F^{\mathsf{#1}}(#2)}
\newcommand{\snet}[2]{\mathcal S^{\mathsf{#1}}(#2)}
\newcommand{\znet}[2]{\mathcal S^{\mathsf{#1}}_{\mathrm Z}(#2)}
\newcommand{\dnet}[2]{\mathcal S^{\mathsf{#1}}_{\mathrm D}(#2)}
\newcommand{\sfA}{\textsf{A}}
\newcommand{\msfA}{\mathsf{A}}
\newcommand{\sfBB}{\textsf{B}}
\newcommand{\msfBB}{\mathsf{B}}
\newcommand{\sfC}{\textsf{C}}
\newcommand{\msfC}{\mathsf{C}}
\newcommand{\sfBC}{\textsf{BC}}
\newcommand{\msfBC}{\mathsf{BC}}
\newcommand{\YBCq}{Y_q^\mathsf{BC}}
\newcommand{\vertex}[1]{\mathsf{#1}}
%\renewcommand{\st}{^{\text{st}}}
%\renewcommand{\nd}{^{\text{nd}}}
%\renewcommand{\rd}{^{\text{rd}}}
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\begin{document}

\maketitle
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\section{Introduction}\label{s:intro}

%\begin{enumerate}
%\item ( Define totally nonnegative matrices)
Given an
%real 
$n\times n$ matrix $A = (a_{i,j})$
and subsets $I, J \subseteq [n] \defeq \{1,\dotsc, n\}$,
let $A_{I,J} = (a_{i,j})_{i \in I, j \in J}$ denote the $(I,J)$-{\em submatrix} of $A$.
For $|I|= |J|$, call $\det(A_{I,J})$ the $(I,J)$-{\em minor} of $A$.
A real $n\times n$ matrix $A$ is called \emph{totally positive} ({\em totally nonnegative}) if every minor of $A$ is positive (nonnegative). Let $\tp n \subset \tnn n$ denote these sets of matrices.
%these
% Initially, total positivity arose in three different areas. It was studied by Gantmacher and Krein in small oscillations of vibrating systems (see \cite{GKOsz},\cite{GKOsc}), by Sch\"onberg in applications to analysis of real roots of polynomials and spline functions (see \cite{cur}), and by Karlin in integral equations and statistics (see \cite{ando87}, \cite{gas2013total}). 
%Totally positive 

These and the set $\hps n$ of $n \times n$ Hermitian positive semidefinite matrices
%Such matrices 
arise in many areas of mathematics,
%. (See, e.g., \cite{FJTNNMatrices}).
%$\quad$ 
% For $I,I' \subseteq \{1,2,\ldots,n\}$ with
% $|I|=|I'|$, we denote the permanent of a submatrix of  $A$ with row set $I$ and column
% set $I'$ as $\text{per}(A_{I,I'})$.
       % \item Mention long history of studying inequalities in these
    % matrices and in HPSD matrices, especially concerning products of determinants and permanents of submatrices.  Include a few of the examples from our last introduction?
and for more than a century mathematicians have been studying inequalities satisfied by their matrix entries. 
(See, e.g., \cite{FJTNNMatrices}.)
%of totally nonnegative and Hermitian positive semidefinite matrices.  (See, e.g., \cite{FJTNNMatrices}.) 
Many such inequalities involve minors and permanents.
%\begin{equation*}
%    \perm(A) \defeq \sum_{w \in \sn} \permmon aw.
%\end{equation*}
For instance
inequalities of Fischer~\cite{fischer1908}, 
Fan~\cite{carlson1967},
and Lieb~\cite{LiebPerm}
state that 
 for all matrices $A \in \tnn n \cup \hps n$, 
 %the above classes 
 %of matrices 
 and for all $I \subseteq [n]$
 %$ \defeq \{ 1,\dotsc, n\}$ 
 and $I^c \defeq [n] \ssm I$, we have
    \begin{equation}\label{eq:fischer}
    \begin{gathered}
    \det(A)\leq \det(A_{I,I}) \det(A_{I^c,I^c}),\\
    \perm(A) \geq \perm(A_{I,I})\; \perm(A_{I^c,I^c}).
    \end{gathered}
    \end{equation}
%     and Ky Fan showed that this holds for $A$ TNN as well (unpublished; 
%(see \cite{carlson1967}).
%%Lieb~\cite{LiebPerm} proved a permanental analog
    % \begin{equation}\label{eq:lieb}
    % \perm(A)\geq \perm(A_{I,I}) \;\perm(A_{I^c,I^c}),
    % \end{equation}
    % for $A$ HPSD, and this analog holds for $A$ TNN because every term in the expansion of the product on the right-hand side of (\ref{eq:lieb}) also appears on the left-hand side.
Koteljanskii's inequality~\cite{KotelProp}
%, \cite{KotelPropRuss} 
states that for $A \in \tnn n \cup \hps n$ and %for all 
$I, J \subseteq [n]$ %with $|I| = |J|$ 
we have
\begin{equation}\label{eq:koteljanskii}
    \det(A_{I\cup J, I\cup J})\det(A_{I\cap J, I\cap J})\leq \det(A_{I,I}) \det(A_{J,J}).
    \end{equation}
%    for $A$ belonging to a class of matrices including HPSD and TNN matrices ~\cite{KotelProp}, \cite{KotelPropRuss}.  

Many open questions about inequalities exist and seem difficult. For instance, it is known which $8$-tuples 
$(I,J,K,L,I',J',K',L')$ of subsets satisfy
\begin{equation}
    \det(A_{I,I'})\det(A_{J,J'}) \leq \det(A_{K,K'})\det(A_{L,L'}) 
\end{equation}
for all 
%totally nonnegative 
$A \in \tnn n$ \cite{FGJMult}, \cite{SkanIneq}, but few permanental analogs of such inequalities are known.  While some of these $8$-tuples also satisfiy
%but it is not known which of these $8$-tuples satisfy 
\begin{equation}
    \perm(A_{I,I'})\;\perm(A_{J,J'}) \geq \perm(A_{K,K'})\;\perm(A_{L,L'}),
\end{equation}    
    this second inequality is not true in general:
%    For example, 
the 
%Optionally, assume that the submatrices are principal and complementary within products: 
%$\perm(A_{I,I}) \perm(A_{I^c,I^c}) \leq \perm( A_{K,K}) \perm(A_{K^c,K^c})$, etc.
%$\det(A_{I,I}) \det(A_{I^c,I^c}) \leq \det( A_{K,K}) \det(A_{K^c,K^c})$.
 natural permanental analog
 %Koteljanskii's inequality 
\begin{equation}\label{eq:kotelperm}
\perm(A_{I\cup J, I\cup J})\;\perm(A_{I\cap J, I\cap J})\geq \perm(A_{I,I})\; \perm(A_{J,J})
\end{equation}
%for all $I, J \subseteq [n]$.
%Somewhat surprisingly, this inequality 
of \eqref{eq:koteljanskii}
holds neither for all
$A \in \hps n$ nor for all $A \in \tnn n$.
%Hermitian positive semidefinite matrices nor for totally nonnegative matrices. 
(See \cite[Section 6]{SkanSoskinBJArx} for a counterexample with $n = 3$.)

%We will not consider inequalities for elements of $\hps n$ any further,
%but 
Let us put aside $\hps n$ and consider 
%the study of 
conjectured inequalities of the form
%One strategy for studying a conjectured inequality
%of the above forms, i.e., 
\begin{equation}\label{eq:productineq}
\mathrm{product}_1 \leq \mathrm{product}_2
\end{equation}
%for 
%satisfied by 
%products of
involving minors and permanents of matrices in $\tnn n$
and $\tp n$.
%such as (\ref{eq:fischer}) -- (\ref{eq:kotelperm})
%those above
%for $\tnn n$
One strategy for studying \eqref{eq:productineq} is to 
%subtract 
%one product from the other 
%and 
view the difference $\mathrm{product}_2 - \mathrm{product}_1$ as a polynomial 
\begin{equation}\label{eq:poly}
    f(x) \defeq f(x_{1,1}, x_{1,2}, \dotsc, x_{n,n}) \in \zx \defeq \zxn
\end{equation}
in matrix entries.  Then the validity of the inequality \eqref{eq:productineq} is equivalent to the statement that for all $A = (a_{i,j}) \in \tnn n$, we have
\begin{equation}\label{eq:tnnpoly}
    f(A) \defeq f(a_{1,1}, a_{1,2}, \dotsc, a_{n,n}) \geq 0.
\end{equation}
We call a polynomial \eqref{eq:poly} with this property
a {\em totally nonnegative polynomial}.
Since $\tp n$ is dense in $\tnn n$, the inequality \eqref{eq:tnnpoly} holds for all $A \in \tp n$ if and only if it holds for all $A \in \tnn n$.

A second strategy 
%for studying inequalities such as 
for studying (variations of) a potential inequality \eqref{eq:productineq}
%such as (\ref{eq:fischer}) -- (\ref{eq:kotelperm})
%those above
%for $\tnn n$
%those above 
is to ask for which positive constants $k_1$, $k_2$
the modified inequalities
\begin{equation}\label{eq:modineq}
k_1 \cdot \mathrm{product}_1 \leq \mathrm{product}_2 
%\qquad
%\mathrm{product}_2 
\leq k_2 \cdot \mathrm{product}_1
\end{equation}
%multiply one side by an unknown
%positive constant 
%$k$ and to ask for which $k$ the modified inequality 
hold for all $A \in \tnn n$.  
Bounds of $k_1 = 1$ or $k_2 = 1$ imply
the inequality \eqref{eq:productineq} or its reverse to hold; %respectively; 
other bounds give information not apparent in the proof or disproof of \eqref{eq:productineq}.
Equivalently, we may view the ratio of $\mathrm{product}_2$ to $\mathrm{product}_1$
%of the two sides of 
%(\ref{eq:productineq})
%the inequality 
as a rational function
\begin{equation}\label{eq:origratio}
    R(x) \defeq R(x_{1,1}, x_{1,2}, \dotsc, x_{n,n}) \in \mathbb Q(x) \defeq \mathbb Q(x_{1,1}, x_{1,2}, \dotsc, x_{n,n})
\end{equation}
in matrix entries, and we may ask for upper and lower bounds
as $x$ varies over $\tp n$.
While a ratio \eqref{eq:origratio}  
%Equivalently, we may state upper and lower bounds for the ratio of the two sides of the inequality, considered as a real valued function $R(x)$ on the set of totally nonnegative matrices $x=(x_{ij})$.
%It is also clear that some such ratios 
is not defined everywhere on $\tnn n$, 
%since matrix entries and other minors may be $0$,
%Again, 
the density of $\tp n$ in $\tnn n$ allows us to
%on the set of totally nonnegative matrices, whose entries and other minors may be zero.  We will therefore 
restrict our attention to $\tp n$:
%the set $\tp n$ of $n \times n$ totally positive matrices.  Since the set $\tp n$ is dense in the set $\tnn n$ of $n \times n$ totally nonnegative matrices, bounds for the above ratios apply also to those $x \in \tnn n$ 
%totally nonnegative matrices 
%for which they are defined,
we have
\begin{equation}\label{eq:bounds}
k_1 \leq R(x) \leq k_2
\end{equation}
for all $x \in \tp n$ if and only if
the same inequalities hold for all 
$x \in \tnn n$ such that $R(x)$ is defined.
% \begin{equation}\label{eq:bounds}
%      \begin{aligned}
%          %0 < 
%          k_1 &\leq R(x) \leq k_2 \quad \text{for all $x \in \tp n$} \qquad \qquad \Longleftrightarrow\\
% %         &\text{if and only if} \\
%          %0 < 
%          k_1 &\leq R(x) \leq k_2 \quad \text{for all $x \in \tnn n$ such that $R(x)$ is defined.}
%      \end{aligned}
%  \end{equation}
Clearly 
the lower bound $k_1$ is interesting
%for $R(x)$ are interesting 
%only if nontrivial, i.e., positive, since 
%Clearly 
only when positive,
since 
%products of 
minors and permanents of totally nonnegative matrices are trivially bounded below by $0$.
%Furthermore, bounds of $k_1 = 1$ or $k_2 = 1$ imply
%the inequality (\ref{eq:productineq}) or its reverse to hold, %respectively.

%is always bounded below by zero.

%It is also clear that 

% \begin{equation}\label{eq:bounds}
%     \begin{gathered}
%         0 < k_1 \leq R(x) \leq k_2 \quad \text{for all $x \in \tp n$}\\
%  \Longleftrightarrow \\
%        \text{if and only if} \\
%         0 < k_1 \leq R(x) \leq k_2 \quad \text{for all $x \in \tnn n$ such that $R(x)$ is defined.}
%     \end{gathered}
% \end{equation}
%We will not consider Hermitian positive semidefinite matrices.

%We proceed to consider inequalities from the point of view of (\ref{eq:origratio}) and the second strategy above.
%in order to include pairs of products of minors and permanents which appear incomparable
%from the point of view of the first strategy.
A characterization of all ratios of the form
\begin{equation}\label{eq:2over2}
\frac{\det(x_{I,I'})\det(x_{J,J'})}
{\det(x_{K,K'})\det(x_{L,L'})}, \qquad I, I',\dotsc, L, L' \subset [n],
\end{equation}
which are bounded above and/or nontrivially bounded below on $\tp n$
follows from work in
\cite{FGJMult}, \cite{SkanIneq}.
%were characterized.  
Each ratio \eqref{eq:2over2} is bounded above and/or below by $1$, and for each $n$, 
%Specifically, for each $n$,
%each bounded ratio (\ref{eq:2over2}) factors 
%these 
it factors as a product of elements of a finite set of indecomposable ratios.  This result was extended in \cite{GSBoundedRatios} to include ratios of products of arbitrarily many minors
\begin{equation}\label{eq:manyovermany}
\frac{\det(x_{I_1,I'_1}) \cdots \det(x_{I_p,I'_p})}
{\det(x_{J_1,J'_1})\cdots\det(x_{J_p,J'_p})}.
\end{equation}
Again, each such ratio
%of these 
factors
%Again, for each $n$, each bounded ratio (\ref{eq:manyovermany}) factors 
as a product of elements belonging to a finite set of indecomposable ratios.  For $n = 3$, each ratio \eqref{eq:manyovermany} is bounded above and/or below by $1$;
%the upper and/or lower bounds are known to be $1$; 
for $n \geq 4$, such bounds are conjectured~\cite{BoochGen}.





%In all cases, the ratios (\ref{eq:2over2}) are bounded by $1$.

%Fallat--Gekhtman--Johnson used this point of view in \cite{} and found

 %   $\star$ Mention that Charlie Johnson and others studied polynomial inequalities from the rational function point of view.  Mention implications between the two points of view.  
 While the permanental version \eqref{eq:kotelperm} of Koteljanskii's inequality is false, we will show in \Cref{s:main} that the corresponding ratio is bounded above and nontrivially below.
 Specifically,
%There is a 
%From the results in this paper we show that 
\begin{equation}\label{eq:kotelratio}
\frac1{|I \cup J|! ~| I \cap J|!}
\leq
\frac{\perm(x_{I,I})\;\perm(x_{J,J})}
{\perm(x_{I\cup J, I\cup J})\;\perm(x_{I\cap J, I\cap J})}
\leq
|I|!~|J|!
\end{equation}
%\geq \frac{\perm(A_{I,I})\; \perm(A_{J,J})}{|I|!~|J|!}
%\end{equation}
for all $I, J \subseteq [n]$ and $x \in \tp n$.
The 
%combination of (\ref{eq:kotelratio}) and the 
failure of \eqref{eq:kotelperm}, combined with \eqref{eq:kotelratio},
%shows that unlike the bounded ratios of minors
exposes a difference between ratios of minors and of permanents:
%implies
%it is clear 
unlike the bounded ratios in \eqref{eq:2over2},
%that 
{\em not} all bounded ratios of permanents are bounded by $1$.  Thus it is natural to ask
which ratios 
\begin{equation}\label{eq:R}
R(x)=\frac{\text{per}(x_{I_1,I'_1})\text{per}(x_{I_2,I'_2})\cdots\text{per}(x_{I_r,I'_r})}{\text{per}(x_{J_1,J'_1})\text{per}(x_{J_2,J'_2})\cdots\text{per}(x_{J_q,J'_q})}
\end{equation} 
%of permanents 
are bounded above and/or nontrivially below as real-valued functions on $\tp n$,
and to look for bounds.
 %in conditions on  $R$ to be bounded 
% as a real-valued function on the locus of totally positive elements in $GL_{n}$, and to state bounds where possible. 

In \Cref{s:multitnnorder} we describe
%look closely at
%monomials in 
a multigrading of 
the coordinate ring $\zx$ of $n \times n$ matrices.  Extending work in \cite{DGSBruhat}, we define a partial order on the monomials in 
%each multigraded component of 
$\zx$ which characterizes the differences 
$\smash{\prod x_{i,j}^{c_{i,j}} - \prod x_{i,j}^{d_{i,j}}}$ 
%of two monomials 
which are totally nonnegative polynomials.
%facilitates the 
%study of total nonnegativity.
%define a partial order on these
%which aids in the study of total nonnegativity and which %extends a partial order  
%use total nonnegativity to  extend the partial order 
%defined in \cite{DGSBruhat}.
%to all monomials in this ring.  
This leads to our main results in \Cref{s:main} which characterize ratios \eqref{eq:R} which are bounded above and nontrivially below as real-valued functions on $\tp n$.  We provide some such bounds, which are not necessarily tight, and suggest problems for further investigation.
%the locus of totally positive $n \times n$ matrices.
%We finish in \Cref{s:open} with some open questions.
%\begin{equation}\label{eq:kotelperm}
%\frac{\perm(A_{I\cup J, I\cup J})\;\perm(A_{I\cap J, I\cap J})}{|I\cup J|!~|I\cap J|!}\leq \perm(A_{I,I})\; \perm(A_{J,J})
%\end{equation}
%for all $I, J \subseteq [n]$.

    
%     $\star$ Mention what is known about bounded ratios of minors, and what questions are open.


    
    
% Determinantal inequality is referred to as multiplicative when it compares two products of minors. The problem of description of multiplicative determinantal inequalities was stated by S. Fallat and C. Johnson in \cite{fallat2000det} in terms of bounded ratios of product of minors. In \cite{fallat2003multiplicative} a large class of bounded ratios
% of products of principal minors was classified. In particular, necessary
% and sufficient conditions were stated for a ratio of products of two minors to
% be bounded over totally positive matrices.

% This result was generalized to non-principal minors in
% \cite{skandera2004inequalities}.
% Necessary condition in
% \cite{fallat2003multiplicative} was generalized in \cite{boocher2008generators} to the case of non-principal minors and an explicit factorization of ratios of products of two minors
% into products of primitive ratios was constructed.
% It has been conjectured that all bounded ratios can be factored into products
% of \emph{primitive ratios}
% \cite{fallat2003multiplicative}. Recently, it was proved in \cite{} that for any $n\times n$ matrix the set of bounded ratios is finitely generated. Along with that some examples of nonprimitive generators were discovered which disprove conjecture on factorization into primitive ratios stated in \cite{fallat2003multiplicative}.
%     $\star$ Mention that it is a natural to ask the same questions about 
%     products of permanents.  Mention that while many permanental analogs
%     of determinantal facts are more difficult, we will show that in this
%     case, the answers and proofs are in fact easier.
%     $\star$ Outline paper, saying in Section 2 we will do X, in Section 3 we will do Y, etc.
%  %   \end{enumerate}

    
% %\end{enumerate}

% %$\star$ Focus the introduction on bounded ratios of minors.  Mention results of Misha and others.

% %$\star$ Raise the question of which
% ratios of permanents are bounded.
% %\\
% %\\



% The following inequalities hold for  
% %$n \times n$ 
% %matrices $A$ which are 
% HPSD (hermitian positive semidefinite) matrices and TNN matrices.
% %matrices $A$ of size $n \times n$.
% Hadamard~\cite{hadamard1893} showed that for $A$ HPSD we have
% %\begin{enumerate}
% %\item 
% \begin{equation}\label{eq:hadamard}
%     \det(A)\leq a_{1,1} \cdots a_{n,n},
%     \end{equation}
% and     Koteljanskii~\cite{KotelProp}, \cite{KotelPropRuss} showed that this holds for $A$ TNN as well.
%     Marcus~\cite{marcus1963perm} proved a permanental analog
%     \begin{equation}\label{eq:marcus}
%     \perm(A)\geq a_{1,1} \cdots a_{n,n}
%     \end{equation}
%     for $A$ HPSD, and this analog clearly holds for $A$ TNN as well.
%     Fischer~\cite{fischer1908} strengthened (\ref{eq:hadamard}) by showing that for all $I \subseteq [n]$ we have
%     \begin{equation}\label{eq:fischer}
%     \det(A)\leq \det(A_{I,I}) \det(A_{I^c,I^c}),
%     \end{equation}
%      and Ky Fan showed that this holds for $A$ TNN as well (unpublished; see \cite{carlson1967}).
%     Lieb~\cite{LiebPerm} proved a permanental analog
%     \begin{equation}\label{eq:lieb}
%     \perm(A)\geq \perm(A_{I,I}) \;\perm(A_{I^c,I^c}),
%     \end{equation}
%     for $A$ HPSD, and this analog holds for $A$ TNN because every term in the expansion of the product on the right-hand side of (\ref{eq:lieb}) also appears on the left-hand side.
%     Koteljanskii~\cite{KotelProp}, \cite{KotelPropRuss} strengthened (\ref{eq:fischer}) further by proving that for all $I, J \subseteq [n]$ we have
%     \begin{equation}\label{eq:koteljanskii}
%     \det(A_{I\cup J, I\cup J})\det(A_{I\cap J, I\cap J})\leq \det(A_{I,I}) \det(A_{J,J})
%     \end{equation}
%     for $A$ belonging to a class of matrices including HPSD and TNN matrices.  
%     %No permanental analog of (\ref{eq:koteljanskii}) is known.
    
    
% %    The permanental 
    
% Other inequalities hold for 
% %$n \times n$ 
% TNN matrices, and are not known to hold for HPSD matrices.
% For instance Charles Johnson showed that
% \begin{equation}\label{eq:permdominance}
% \perm(A) \geq \frac{\imm{\chi}(A)}{\chi(e)},
% \end{equation}
% holds for TNN matrices (unpublished; see \cite[p.\,1088]{StemConj}).  The validity of this inequality for HPSD matrices was conjectured by Lieb~\cite{LiebPerm}, is still open, and is known as 
% the {\em permanental dominance conjecture}.
% %~\cite{LiebPerm}.
% %is open for HPSD matrices. Johnson (unpublished but known for TNN matrices.
% Extending the results (\ref{eq:fischer}) and (\ref{eq:koteljanskii}), Fallat, Gekhtman, and Johnson~\cite{FGJMult}
% and the first author~\cite{SkanIneq} characterized all $8$-tuples
% $(I,J,K,L,I',J',K',L')$ of subsets of $[n]$
% for which we have the inequality
% %inequalities of the form
% %quadruples $(I,J,K,L)$ of subsets of $[n]$ for which we have the inequality of the form
% \begin{equation}\label{eq:fgjskan}
% \det(A_{I,I'})\det(A_{J,J'}) \leq 
% \det(A_{K,K'})\det(A_{L,L'})
% \end{equation}
% whenever $A$ is TNN.
% (See \cite{GkhtSos2022bounded} for progress on products of three or more minors.) 
% %While no characterization is known for products
% %of arbitrarily many minors at a time,
% Drake, Gerrish, and the first author~\cite{DGSBruhat} proved a similar result for products of $n$ matrix entries.
% %minors of size $1 \times 1$.  
% In particular, the inequality
% \begin{equation}\label{eq:dgs}
%     \permmon av \geq \permmon aw
% \end{equation}
% holds for all TNN $A$ if and only if $v \leq w$ in the Bruhat order on $\sn$. (See, e.g., \cite{BBCoxeter}.)
% %where $I,J,K,L,I',J',K',L' \subseteq [n]$.
% %This characterization was extended in \cite{SkanIneq} to include nonprincipal minors, i.e., 
% %\begin{equation}\label{eq:skan}
% %\det(A_{I,I'})\det(A_{J,J'}) \leq 
% %\det(A_{K,K'})\det(A_{L,L'}),
% %\end{equation}
% %where $I,J,K,L,I',J',K',L' \subseteq [n]$.
% %i.e., \det(A_{I,I'}
% %\begin{enumerate}
% %    \item For certain $I,J,K,L, I', J', K', L' \subseteq [n]$, we have  \begin{equation*}
% %        \det(A_{I,I'})\det(A_{J,J'}) - \det(A_{K,K'}) \det(A_{L,L'}) \geq 0.\end{equation*} (Fallat, Gekhtman, Johnson , Skandera \cite{SkanIneq}),
%      %Rhoades, Skandera \cite{RSkanTLImm})
%  %    \item 
% % For $v, w \in \sn$ we have $\permmon xv \geq \permmon xw$ if and only if $v \leq w$ in the Bruhat order (Drake, Gerrish, Skandera~\cite{DGSBruhat}, \cite{DGSMNNBruhat}).
%  % \item 
%  Other linear combinations of the monomials in (\ref{eq:dgs}) which produce valid inequalities for TNN matrices are related to Lusztig's work on canonical bases of quantum groups~\cite{LusztigTPCB} and are called \emph{Kazhdan--Lusztig immanants} in \cite{RSkanKLImm}.
%  Writing these as
%  $\{ \imm{w}(A) \,|\, w \in \sn \}$, we have
%  \begin{equation}\label{eq:klimm}
%  \imm w(A) \geq 0
%  \end{equation}
% for all $w \in \sn$ and all TNN $A$.
% (See \cite{RSkanKLImm} for definitions.)
 
% We are interested in ratios of products of permanents of submatrices $A$
% \begin{equation}\label{eq:R}
% R(A)=\frac{\text{per}(A_{I_1,I'_1})\text{per}(A_{I_2,I'_2})...\text{per}(A_{I_r,I'_r})}{\text{per}(A_{J_1,J'_1})\text{per}(A_{J_2,J'_2})...\text{per}(A_{J_q,J'_q})},
% \end{equation}  
% more precisely, in conditions on  $R$ to be bounded as a real-valued function on the locus of totally positive elements in $GL_{n}$.

% In \Cref{s:multitnnorder} we discuss a multigrading of the coordinate ring $\zxn$ of $n \times n$ matrices, and use total nonnegativity to  extend the partial order defined in \cite{DGSBruhat} to all monomials in this ring.  This leads to our main results in \Cref{s:main} which characterize ratios (\ref{eq:R}) which are bounded as real-valued functions on the locus of totally positive $n \times n$ matrices.
% We finish in \Cref{s:open} with some open questions.
\section{
%The total nonnegativity order}
A multigrading of \texorpdfstring{$\zxn$}{Z[x11,x12,...,xnn]} and the total nonnegativity order}
\label{s:multitnnorder}

% Many inequalities satisfied by the entries of totally nonnegative matrices
% can be expressed in terms of polynomials in the ring $\zx \defeq
% \zxn$, where we view $x = (x_{i,j})$ as a matrix of indeterminates.
% In particular, for $f(x) \in \zx$ and $A = (a_{i,j})$ an $n \times n$ matrix,
% we define
% we might express an inequality
% in the form
% \begin{equation}\label{eq:tnnpolydef}
%   f(A) \defeq f(a_{1,1}, a_{1,2}, \dotsc, a_{n,n}) \geq 0
%   \text{ for all totally nonnegative matrices } A.
% \end{equation}
% In this case, we call $f(x)$ a {\em totally nonnegative polynomial}.
% ($\star$ Move this definition to the introduction?)


We will find it convenient to view degree-$r$ monomials in $\zx$
in terms of permutations in the symmetric group $\sr$ and multisets of
$[n]$.
%$ \defeq \{1, \dotsc, n\}$.
In particular, given permutations $v, w \in \sr$ define
the monomial
\begin{equation*}
    x^{v,w} \defeq x_{v_1,w_1} \cdots x_{v_r,w_r}.
\end{equation*}
Define an {\em $r$-element multiset} of $[n]$ to be a nondecreasing
$r$-tuple of elements of $[n]$.
In exponential notation, we write $i^{k}$ to represent $k$ consecutive
occurrences of $i$ in such an $r$-tuple, e.g.,
\begin{equation}\label{eq:multisetex}
    (1,1,2,3) = 1^22^13^1, \qquad (1,2,2,2) = 1^12^3.
\end{equation}
Two $r$-element multisets
\begin{equation}\label{eq:MO}
%    \begin{aligned}
%        M &= (m_1,\dotsc,m_r) = 1^{\alpha_1} \cdots n^{\alpha_n},\\
%        O &= (o_1,\dotsc,o_r) = 1^{\beta_1} \cdots n^{\beta_n},
%    \end{aligned}
M = (m_1,\dotsc,m_r) = 1^{\alpha_1} \cdots n^{\alpha_n}, \qquad
O = (o_1,\dotsc,o_r) = 1^{\beta_1} \cdots n^{\beta_n},
\end{equation}
determine a {\em generalized submatrix}
$x_{M,O}$ of $x$ by $(x_{M,O})_{i,j} \defeq x_{m_i,o_j}$.
For example, when $n=3$, we have the $4 \times 4$ generalized submatrix
and monomial
\begin{equation}\label{eq:gensubmatrixex}
    x_{1123,1222} = \begin{bmatrix}
        x_{1,1} & x_{1,2} & x_{1,2} & x_{1,2} \\
        x_{1,1} & x_{1,2} & x_{1,2} & x_{1,2} \\
        x_{2,1} & x_{2,2} & x_{2,2} & x_{2,2} \\
        x_{3,1} & x_{3,2} & x_{3,2} & x_{3,2}
    \end{bmatrix}\ntksp,
    \qquad
    (x_{1123,1222})^{1234,4312} = x_{1,2} x_{1,2} x_{2,1} x_{3,2}.
\end{equation}

The ring $\zx$ has a natural multigrading
\begin{equation}\label{eq:multigrading}
  \mathbb Z[x] = \bigoplus_{r \geq 0} \bigoplus_{M,O} \mathcal A_{M,O},
\end{equation}
where the second direct sum is over pairs $(M,O)$ of $r$-element
multisets of $[n]$, and
\begin{equation}\label{eq:permmon}
  \mathcal A_{M,O} \defeq
  \spn_{\mathbb Z}\{ (x_{M,O})^{e,w} \,|\, w \in \sr \}.
  \end{equation}
More precisely, for $M$, $O$ as in \eqref{eq:MO},
a basis for $\mathcal A_{M,O}$ is given by all monomials
\begin{equation}\label{eq:multimon}
  \prod_{i,j \in [n]} \ntksp x_{i,j}^{c_{i,j}}
  \end{equation}
%x_{1,1}^{c_{1,1}} x_{1,2}^{c_{1,2}} \cdots x_{n,n}^{c_{n,n}}$
with $C = (c_{i,j}) \in \mathrm{Mat}_{n\times n}(\mathbb N)$ satisfying
\begin{equation}\label{eq:CtoMO}
%  \begin{gathered}
%    C = (c_{i,j}) \in \mat n(\mathbb N),\\
    c_{i,1} + \cdots + c_{i,n} = \alpha_i,
    %d_{i,1} + \cdots + d_{i,n}
    \quad
%    \text{for } i=1,\dotsc,n, \qquad
    c_{1,j} + \cdots + c_{n,j} = \beta_j
%    d_{1,j} + \cdots + d_{n,j}
    \quad \text{for } i,j=1,\dotsc,n.
%    \end{gathered}
%  \end{aligned}
  \end{equation}
%  where $\alpha_i$ and $\beta_j$ are defined as in (\ref{eq:MO}).
One may express a monomial \eqref{eq:multimon}
in the form $(x_{M,O})^{e,w}$
%(\ref{eq:permmon})
%$x_{M,O}^{e,w}$
by the following algorithm.
\begin{alg}\label{a:mattoperm}
  Given a monomial \eqref{eq:multimon}
  %of degree $r$
  in $\mathcal A_{M,O}$
  with $M$, $O$ as in \eqref{eq:MO},
  \begin{enumerate}[label=(\roman*)]
    \item Define the rearrangement $u = u_1 \cdots u_r$ of $O$ by
  writing \eqref{eq:multimon} with variables in lexicographic order
  as $x_{m_1,u_1} \cdots x_{m_r,u_r}$.
\item Let $j_1 < \cdots < j_{\beta_1}$ be
  the positions of the $\beta_1$ ones in $u$,
  let $j_{\beta_1 + 1} < \cdots < j_{\beta_1 + \beta_2}$ be
  the positions of the $\beta_2$ twos in $u$, etc.
\item For $i = 1, \dotsc, r$, define $w_{j_i} = i$.
  %place letter $i$ in position $j_i$ of $w$.
\item Call the resulting word $w = w(C)$.
\end{enumerate}
\end{alg}
  %and exponent matrix $C$ (\ref{eq:CtoMO}).
%    $M = (m_1,\dotsc,m_r) = 1^{\alpha_1} \cdots n^{\alpha_n}$,
%     (1^22^13^1, 1^12^33^0)$ of $\{1,2,3\}$,
For example, it is easy to check that for 
%$n=3$ and 
multisets
$(1123,1222) = (1^22^13^1, 1^12^33^0)$ of $\{1,2,3\}$,
the graded component $\mathcal A_{1123,1222}$
%= \mathcal A_{1^22^13^1,1^12^33^0}$
of $\mathbb Z[x_{1,1},x_{1,2},\dotsc,x_{3,3}]$
is spanned by monomials \eqref{eq:multimon},
where $C = (c_{i,j})$ is one of the matrices
%the $3 \times 3$ matrices
\begin{equation}\label{eq:3x3matrices}
  \begin{bmatrix}
    1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0
  \end{bmatrix}\ntksp,
  \quad
  \begin{bmatrix}
    0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0
  \end{bmatrix}\ntksp,
  \quad
  \begin{bmatrix}
    0 & 2 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0
  \end{bmatrix}
  %\ntksp.
\end{equation}
having row sums $(2,1,1)$ and column sums $(1,3,0)$.  These monomials are
\begin{equation}\label{eq:3monomials}
%  \begin{gathered}
   x_{1,1}x_{1,2}x_{2,2}x_{3,2}, \quad
   x_{\smash{1,2}}^2x_{2,1}x_{3,2}, \quad
   \smash{x_{1,2}^2}x_{2,2}x_{3,1},
%  \end{gathered}
\end{equation}
with column index sequences equal to the rearrangements
$1222$, $2212$, $2221$ of $1222$.
\Cref{a:mattoperm} then produces
permutations $1234$, $2314$, $2341$ in $\mfs 4$, and we may express
the monomials \eqref{eq:3monomials} as
\begin{equation}\label{eq:3moremonomials}
%  \mathcal A_{1123,1222} = \spn
  %\{
  (x_{1123,1222})^{1234,1234},\quad
  (x_{1123,1222})^{1234,2314},\quad
  (x_{1123,1222})^{1234,2341}.
  %\}.
\end{equation}

%Applying Algorithm~\ref{a:mattoperm} to produce basis elements
%of the graded component $\mathcal A_{1123,1222}$ of
%$\mathbb Z[x_{1,1},x_{1,2},\dotsc,x_{3,3}]$,
%we obtain
%\begin{equation*}
%  \begin{gathered}
%   x_{1,1}x_{1,2}x_{2,2}x_{3,2} = (x_{1123,1222})^{1234,1234},\\
%  x_{\smash{1,2}}^2x_{2,1}x_{3,2} = (x_{1123,1222})^{1234,3124},\\
%  \smash{x_{1,2}^2}x_{2,2}x_{3,1} = (x_{1123,1222})^{1234,4123}.
%  \end{gathered}
%\end{equation*}
%($\star$ Fix the above.)

  %The total %nonnegativity poset on
\ntnsp For $r$-element multisets $M, \ntnsp O$ of $[n]$,
%$\{1,\dotsc,n\}$, 
%the 
monomials
in $\mathcal A_{M,O}$ are closely
related to parabolic subgroups of $\mfs r$ with standard generators
$s_1, \dotsc, s_{r-1}$, and double cosets of the form
%in $\mfs r$.
%($\star$ Introduce generators of $\sr$ and parabolic subgroups $W_I$.)
%These are sets of the form
$W_{\iota(M)} w W_{\iota(O)}$
  %$I$, $J$ are subsets of generators of $\mfs k$.
  %isomorphic to the Bruhat order on double cosets
  %$W_{\iota(M)} \backslash W / W_{\iota(O)}$,
where $w$ belongs to
%\in 
$\mfs r$,
$W_J$ is the subgroup of $\mfs r$ generated by $J$, and
  \begin{equation}\label{eq:parabolics}
    \begin{gathered}
      \iota(M) \defeq \{ s_1, \dotsc, s_{r-1} \} \ssm
      \{s_{\alpha_1}, s_{\alpha_1 + \alpha_2}, \dotsc, s_{r-\alpha_n} \}
      = \{ s_j \,|\, m_j = m_{j+1} \},\\
      \iota(O) \defeq \{ s_1, \dotsc, s_{r-1} \} \ssm
      \{s_{\beta_1}, s_{\beta_1 + \beta_2}, \dotsc, s_{r-\beta_n} \}
      = \{ s_j \,|\, o_j = o_{j+1} \}.
    \end{gathered}
  \end{equation}
  %(or something like that - think about $n$ and $k$).
  %Specifically,
  It is easy to see that the map $M \mapsto \iota(M)$ is bijective:
  one recovers $M = 1^{\alpha_1} \cdots n^{\alpha_n}$ from the generators not in $\iota(M)$
  as in \eqref{eq:parabolics}.
  It is known that each double coset has unique minimal and maximal elements
  with respect to the {\em Bruhat order on $\mfs r$}, defined by declaring $v \leq w$ if each reduced expression $s_{i_1} \cdots s_{i_\ell}$ for $w$ contains a subword which is a reduced expression for $v$.  (See, e.g., \cite{BBCoxeter}, \cite{DougInv}.)
  %if $v = v_1 \cdots v_n$ is
  %the longest element of
%  given $W_{\iota(M)}$, then we can recover $M = m_1 \cdots m_r$
%  by $m_i = 1 + |\{}$.
    Let $W_{\iota(M)} \backslash W / W_{\iota(O)}$ denote
the set of all double cosets of $W = \sr$
determined by $r$-element multisets $M$, $O$.
%  It is known that the group $\mfs r$ is a disjoint union of such cosets.
%  We also have the following facts.
\begin{prop}\label{p:MOcosetcorr}
  Fix $r$-element multisets $M$, $O$ as in \eqref{eq:MO}.
    The double cosets
$W_{\iota(M)} \backslash W / W_{\iota(O)}$ satisfy the following.
%    For $r$-element multisets $M$, $O$ of $\{1,\dotsc,n\}$,
%    the subgroups and cosets defined in (\ref{eq:parabolics}) satisfy the
%    following.
\begin{enumerate}[label=(\roman*)]
\item Each double coset
  has a unique Bruhat-minimal element $u$
  satisfying $su > u$ for all $s \in \iota(M)$ and $us > u$ for all
  $s \in \iota(O)$;
  it has a unique Bruhat-maximal element $u'$
  satisfying $su' < u'$ for all $s \in \iota(M)$ and $u's < u'$ for all
  $s \in \iota(O)$.
\item We have $W_{\iota(M)}vW_{\iota(O)} = W_{\iota(M)}wW_{\iota(O)}$
      if and only if $(x_{M,O})^{e,v} = (x_{M,O})^{e,w}$.
\item The cardinality $|W_{\iota(M)} \backslash W / W_{\iota(O)}|$
  %of these double cosets
  is the dimension of $\mathcal A_{M,O}$, equivalently, the number
  of matrices in $\mat n(\mathbb N)$ having row sums $(\alpha_1,\dotsc, \alpha_n)$
  and column sums $(\beta_1,\dotsc,\beta_n)$.
\item Each permutation $w$ produced by
\Cref{a:mattoperm} is the unique Bruhat-minimal element of
its coset $W_{\iota(M)}wW_{\iota(O)}$.
\end{enumerate}
\end{prop}
  \begin{proof}
  Omitted.
%    (i) See \cite{DougInv}.
%
%    \noindent (ii) The dimension of $\mathcal A_{M,O}$ is the cardinality
%    of
%    the set
%    $\{ (x_{M,O})^{e,w} \,|\, w \in \sr \}$.
%    But we have $(x_{M,O})^{e,v} = (x_{M,O})^{e,w}$
%    if and only if when we partition the $r \times r$
%    permutation matrices $P(v)$, $P(w)$ of $v$, $w$ into blocks by drawing bars
%    after rows $\alpha_1, \alpha_1+\alpha_2,\dotsc,r-\alpha_n$ and after columns
%    $\beta_1, \beta_1+\beta_2, \dotsc, r-\beta_n$, the corresponding blocks of $P(v)$ and $P(w)$
%    contain equal numbers of ones.
%    It follows that for fixed $w \in \sr$, the set
%    $\{ v \in \sr \,|\, (x_{M,O})^{e,v} = (x_{M,O})^{e,w} \}$
%    is $W_{\iota(M)}wW_{\iota(O)}$.
%
    % \noindent (iii)  This follows from (ii), where
    % $c_{i,j}$ is the number of ones in block $(i,j)$ of the permutation
    % matrix of any permutation belonging to the double coset.
%
    % \noindent (iv) By Step (i) of the algorithm,
    % subwords $w_{1} \cdots w_{\alpha_1}$, $w_{\alpha_1+1}\cdots w_{\alpha_1+\alpha_2}$,
    % etc., of $w(C)$ are increasing.
    % It follows that for any
    % generator $s \in \iota(M)$ we have $sw > w$.
    % By Step (ii) of the algorithm,
    % letters $1,\dotsc,\beta_1$ appear in increasing order in $w(C)$,
    % as do $\beta_1+1,\dotsc,\beta_1+\beta_2$, etc.
    % It follows that for any
    % generator $w \in \iota(O)$ we have $ws > w$.
  \end{proof}

%  Certain totally nonnegative polynomials belonging to $\mathcal A_{M,O}$ are
%  related to the Bruhat orders on $\mfs r$ and
%  its double cosets
%  $W_{\iota(M)} \backslash W / W_{\iota(O)}$.
  %  To be precise, t
  For any subsets $I$, $J$ of generators of $\sr$,
  the Bruhat order on $\mfs r$ induces a poset structure on
  %$W_{\iota(M)} \backslash W / W_{\iota(O)}$
  $W_I \backslash W / W_J$
  as follows.  We declare
$W_IvW_J \leq W_IwW_J$ if 
the minimal element of $W_IvW_J$ is less than or equal to the minimal element of $W_IwW_J$.
(Equivalently, we may compare maximal or arbitrary elements of the 
%cosets
%satisfy any of the
%three (equivalent) inequalities in the Bruhat order on $\mfs r$
cosets~\cite[Lemma~2.2]{DougInv}.)
%   \begin{enumerate}[(i)]
%       \item The minimal element of $W_IvW_J$ is less than or equal to
% the minimal element of $W_IwW_J$.
%       \item The maximal element of $W_IvW_J$ is less than or equal to
% the maximal element of $W_IwW_J$.
%       \item At least one element of $W_IvW_J$ is less than or equal to
% at least one element of $W_IwW_J$.
%   \end{enumerate}
%  (See \cite{DougInv}.)
  We call this poset the {\em Bruhat order on
  %the double cosets
  $W_I \backslash W / W_J$}.
  Another
  %fourth 
  equivalent inequality can be stated in terms of
  matrices of exponents defined by monomials in $\mathcal A_{M,O}$.
  (See, e.g., \cite{HoSkanDoubleCoset}.)
  Given a matrix $C = (c_{i,j}) \in \mat n(\mathbb N)$, 
  define the matrix
  $C^* = (c^*_{i,j}) \in \mat n(\mathbb N)$
  by
  \begin{equation}\label{eq:cstar}
    c^*_{i,j} =
    %(c_{1,1} + \cdots + c_{1,j})
    %+ (c_{2,1} + \cdots + c_{2,j})
    %+ \cdots
    %+ (c_{i,1} + \cdots + c_{i,j}).
    \text{sum of entries of $C_{[i],[j]}$}.
  \end{equation}
  %$i \times j$ submatrix of $C$}.
%    \end{equation}
  \begin{prop}\label{p:doubleehresmann}
    Fix monomials
    \begin{equation*}
      (x_{M,O})^{e,v} = \prod_{i,j} x_{i,j}^{c_{i,j}},
      \quad
      (x_{M,O})^{e,w} = \prod_{i,j} x_{i,j}^{d_{i,j}},
    \end{equation*}
    in $\mathcal A_{M,O}$
    and define matrices $C^*$, $D^*$ as in \eqref{eq:cstar}.
    Then we have
    %\begin{equation*}
    $W_{\iota(M)}vW_{\iota(O)} \leq W_{\iota(M)}wW_{\iota(O)}$
    in the Bruhat order if and only if $C^* \geq D^*$
    in the componentwise order.
  \end{prop}
%($\star$ Repeat proof from the unpublished \cite{HoSkanDoubleCoset} here?)

  The Bruhat order on $W_{\iota(M)}\backslash W/W_{\iota(O)}$
    %double cosets
    is closely related to certain totally nonnegative polynomials
    in $\mathcal A_{M,O}$.
%  
%  The definition (\ref{eq:tnnpolydef}) of totally nonnegative polynomials
%  suggests defining a partial order $\leq_T$
%  on all monomials in $\zx$ by declaring
%  \begin{equation}
%    (x_{M,O})^{e,v} \leq_T (x_{P,Q})^{e,w} \ \text{ if } \
%    (x_{P,Q})^{e,w} - (x_{M,O})^{e,v} \text{ is a totally nonnegative polynomia%l. }
%  \end{equation}
%  The subposet of $(\zxn, \leq_T)$ induced by monomials
%  $\{x^{e,w} \,|\, w \in \sn \}$ was shown in \cite{DGSBruhat} to be isomorphic
  %  to the (dual of the)
    Indeed, when $M = O = 1\cdots n$, totally nonnegative polynomials of the form
    $x^{e,v} - x^{e,w}$
    are characterized by the Bruhat order on $\sn$~\cite{DGSBruhat}.
%  Bruhat order on $\sn$.
  \begin{thm}\label{t:DGS}
    For $v, w \in \sn$,
    the polynomial $x^{e,v} - x^{e,w}$ is totally nonnegative
%    we have $x^{e,w} \leq_T x^{e,v}$
    if and only if $v \leq w$ in the Bruhat order.
    %on $\sn$.
  \end{thm}

  We will now extend this result to all monomials in $\zx$.
%  The definition (\ref{eq:tnnpolydef}) of totally nonnegative polynomials
  %  suggests defining a partial order $\leq_T$
  Let us define a partial order $\leq_T$
  on all monomials in $\zx$ by declaring
    $(x_{M,O})^{e,v} \leq_T (x_{P,Q})^{e,w}$  if 
    $(x_{P,Q})^{e,w} - (x_{M,O})^{e,v}$ is a totally nonnegative polynomial.
%  \begin{equation}
%    (x_{M,O})^{e,v} \leq_T (x_{P,Q})^{e,w} \ \text{ if } \
%    (x_{P,Q})^{e,w} - (x_{M,O})^{e,v} \text{ is a totally nonnegative polynomial. }
%  \end{equation}
We call this the {\em total nonnegativity order} on monomials in $\zx$.
It is not hard to show that
the total nonnegativity order is a disjoint
union of its restrictions to the multigraded components \eqref{eq:multigrading}
of $\zx$.
%First we observe the following.
%  \begin{prop}
%    Fix multisets $M$,$O$,$P$,$Q$ of $[n]$.  Then the monomials
%    $(x_{M,O}){e,w}$ and $(x_{P,Q}){e,v}$ are incomparable in the total
%    nonnegativity order unless $M=P$ and $O=Q$.
%  \end{prop}
%  \begin{proof}
%    $\star$ Prove this.
%  \end{proof}
\begin{lem}
  Monomials
\begin{equation}\label{eq:monomials}
  \prod_{i,j} \ntnsp x_{i,j}^{c_{i,j}},
  \quad
  \prod_{i,j} \ntnsp x_{i,j}^{d_{i,j}}
\end{equation}
are comparable in the total nonnegativity order
only if they belong to the same multigraded component
%$\mathcal A_{M,O}$
of $\mathbb Z[x]$.
\end{lem}
\begin{proof} Omitted.
\end{proof}
%   For $k,\ell \in [n]$ and
%   $t \in \mathbb R_{\geq 0}$,
%   define the $n \times n$ matrix $E^{k,\ell}(t) = (e_{i,j}^{k,\ell})_{i,j \in [n]}$ by
%   \begin{equation}\label{eq:eklt}
%     e_{i,j}^{k,\ell} = \begin{cases}
%       t &\text{if $i \leq k$ and $j \leq \ell$},\\
%       1 &\text{otherwise}.
%       \end{cases}
%       \end{equation}
% This matrix is totally nonnegative if $t \geq 1$, or $k = n$, or $\ell = n$.
%
% Suppose that the monomials belong to
% components $\mathcal A_{M,O}$ and $\mathcal A_{M',O'}$ of $\zx$,
% with $M$, $O$, as in (\ref{eq:MO}) and
% \begin{equation*}
%   M' = 1^{\alpha_1'} \cdots n^{\alpha_n'}, \qquad O' = 1^{\beta_1'} \cdots n^{\beta_n'}.
% \end{equation*}
% If $M \neq M'$, then let $k \in [n]$ be the least index
% appearing with different multiplicities in the two multisets.
% Evaluating the monomials (\ref{eq:monomials}) at $E^{k,n}(t)$ yields
% $t^{\alpha_1 + \cdots + \alpha_k}$ and $t^{\alpha'_1 + \cdots + \alpha'_k}$.  The difference of
% these can be made positive or negative by choosing $t<1$ or $t>1$.
% On the other hand, if $O \neq O'$, then
% the evalulation of the monomials (\ref{eq:monomials}) at matrices
% of the form $E^{n,\ell}(t)$ leads to a similar conclusion.
%\end{proof}

%i.e., only if
%\begin{equation*}
%  \begin{aligned}
%    c_{i,1} + \cdots + c_{i,n} &= d_{i,1} + \cdots + d_{i,n}
%    \qquad \text{for all } i,\\
%    c_{1,j} + \cdots + c_{n,j} &= d_{1,j} + \cdots + d_{n,j}
%    \qquad \text{for all } j.
%  \end{aligned}
%  \end{equation*}

%  Associate a permutation $u = u(M,O,w) \in \mfs k$
%  to the triple $(M,O,w)$ as follows.
%  \begin{enumerate}
%  \item Write $O = 1^{\alpha_1} \cdots n^{\alpha_n}$.
%  \item Let $j_1 < \cdots < j_{\alpha_1}$ be the positions of the
%    $\alpha_1$ ones in $w$, let $j_{\alpha_1 + 1} < \cdots < j_{\alpha_1 + \alpha_2}$
%    be the positions of the $\alpha_2$ twos in $w$, etc.
%  \item For $i = 1, \dotsc, k$, place letter $i$ in position $j_i$ of $u$.
%  \end{enumerate}

\begin{thm}\label{t:bruhatdiff}
  Fix $r$-element multisets $M = 1^{\alpha_1} \cdots n^{\alpha_n}$, $O = 1^{\beta_1} \cdots n^{\beta_n}$ 
  as in \eqref{eq:MO},
  %  Fix $r$-element multisets $M$, $O$ as above,
  and matrices $C, D \in \mat n(\mathbb N)$ with row and column sums
  $(\alpha_1,\dotsc,\alpha_n)$, $(\beta_1,\dotsc,\beta_n)$, and define the
  polynomial
    \begin{equation*}
    f(x) = \prod_{i,j}\ntnsp x_{i,j}^{c_{i,j}} - \prod_{i,j}\ntnsp x_{i,j}^{d_{i,j}}
%    x_{m_1,v_1} \cdots x_{m_k,v_k}
%    -
%    x_{m_1,w_1} \cdots x_{m_k,w_k}
  \end{equation*}
in $\mathcal A_{M,O}$.
  %matching the multiplicities in (\ref{eq:MO}).
%  as in (\ref{eq:CtoMO}).
  %  such that
%  and monomials
%    $\prod \ntnsp x_{i,j}^{c_{i,j}}$\ntnsp, $\prod \ntnsp x_{i,j}^{d_{i,j}}$ in
  %$\mathcal A_{M,O}$.
  Then the following are equivalent.
%    and let $v$, $w$, be
    %    rearrangements of $O$.
    \begin{enumerate}%[(i)]
      \item $f(x)$
  is totally nonnegative.
\item $C^* \geq D^*$ in the componentwise order.
\item $w(C) \leq w(D)$ in the Bruhat order on $\sr$.
\item $f(x)$ is equal to a sum of products
of the form $\det(x_{I,J})x_{u_1,v_1} \cdots x_{u_{r-2},v_{r-2}}$ in $\mathcal A_{M,O}$
with $|I| = |J| = 2$.
%  \item ($\star$ Include more equivalent conditions?)
  \end{enumerate}
  \end{thm}
  \begin{proof} Omitted.
%     (i $\Rightarrow$ ii)
%     Suppose $C^* \not \geq D^*$ in the componentwise order and
%     let $(k,\ell)$ the the lexicographically least pair
%     satisfying $c^*_{k,\ell} < d^*_{k,\ell}$.
%     Now choose $t > 1$ and evaluate $f(x)$ at the totally nonnegative
%     matrix $E^{k,\ell}(t)$ to obtain $t^{c^*_{k,\ell}} - t^{d^*_{k,\ell}} < 0$.
%     It follows that $f(x)$ is not a totally nonnegative polynomial.
%     %
%     \noindent
%     (ii $\Rightarrow$ iii)  
%     This follows from \Cref{p:doubleehresmann}
%     and the definition of the Bruhat order on double cosets.
% %
%     \noindent
%     (iii $\Rightarrow$ iv)
%     Suppose that $w(C) \leq w(D)$ and let $p = \ell(w(D)) - \ell(w(C))$. Then there exist 
%     a sequence 
%     \begin{equation*}
%         w(C) = y^{(0)} < y^{(1)} < \cdots < y^{(p-1)} < y^{(p)} = w(D)
%     \end{equation*}
%     of permutations and a
%     sequence $( (i_0,j_0), \dotsc, (i_{p-1\ntnsp},j_{p-1}))$
%     %$(t^{(0)}, \dotsc, t^{(p-1)})$
%     of transpositions in $\sr$ such that
%     we have
%     \begin{equation*}
%         y^{(k)} = (i_{k-1\ntnsp},j_{k-1})y^{(k-1)}, \qquad \ell(y^{(k)}) = \ell(y^{(k-1)}) + 1
%     \end{equation*}
%     for $k = 1, \dotsc, p$.  We may thus write $f(x) = (x_{M,O})^{e,w(C)} - (x_{M,O})^{e,w(D)}$ as the telescoping sum
%     \begin{equation*}
%         \Big(\ntnsp(x_{M,O})^{e,y^{(0)}} \ntksp- (x_{M,O})^{e,y^{(1)}}\Big)
%         + \Big(\ntnsp(x_{M,O})^{e,y^{(1)}} \ntksp- (x_{M,O})^{e,y^{(2)}}\Big)
%         + \cdots +
%         \Big(\ntnsp(x_{M,O})^{e,y^{(p-1)}} \ntksp- (x_{M,O})^{e,y^{(p)}}\Big),
%     \end{equation*}
%     where each parenthesized difference either has the desired
%     form 
%     or is $0$.
% %
% \noindent 
% (iv $\Rightarrow$ i)
% A sum of products of minors is a totally nonnegative polynomial.
 \end{proof}

%  ($\star$ Replace the following example with a bigger example, using
%  the three monomials appearing in earlier examples.)

  For example, let us revisit the monomials
  \eqref{eq:3monomials} -- \eqref{eq:3moremonomials}
  in the graded component
  $\mathcal A_{1123,1222}$ of $\mathbb Z[x_{1,1}, x_{1,2},\dotsc,x_{3,3}]$.
    It is easy to see that $1234 < 2314 < 2341$ in the Bruhat order on
    $\mfs 4$ and that the application of \eqref{eq:cstar}
    to the corresponding matrices in \eqref{eq:3x3matrices}
    yields the componentwise comparisons
  \begin{equation}\label{eq:star3x3matrices}
  \begin{bmatrix}
    1 & 2 & 2 \\ 1 & 3 & 3 \\ 1 & 4 & 4
  \end{bmatrix}
  \geq
  \begin{bmatrix}
    0 & 2 & 2 \\ 1 & 3 & 3 \\ 1 & 4 & 4
  \end{bmatrix}
  \geq
  \begin{bmatrix}
    0 & 2 & 2 \\ 0 & 3 & 3 \\ 1 & 4 & 4
  \end{bmatrix} \ntksp.
\end{equation}
  Thus we have $(x_{1123,1222})^{1234,1234} \geq_T(x_{1123,1222})^{1234,2314} \geq_T (x_{1123,1222})^{1234,2341}$, i.e.,
  %the differences
\begin{equation*}
  x_{1,1}x_{1,2}x_{2,2}x_{3,2}
  \ \geq_T\
  x_{\smash{1,2}}^2x_{2,1}x_{3,2}
  %, \quad
  %x_{1,1}x_{1,2}x_{2,2}x_{3,2} -
  \ \geq_T\
  \smash{x_{1,2}^2}x_{2,2}x_{3,1}.
  %, \quad
%  x_{\smash{1,2}}^2x_{2,1}x_{3,2} - \smash{x_{1,2}^2}x_{2,2}x_{3,1}
\end{equation*}
Furthermore, the chain $1234 < 2134 < 2314 < 2341$ 
in the Bruhat order on $\mfs 4$
with 
\begin{equation}
2134 = (1,2)1234, \quad
2314 = (2,3)2134, \quad
2341 = (3,4)2314
\end{equation}
allows us to write
$x_{1,1}x_{1,2}x_{2,2}x_{3,2} -  x_{\smash{1,2}}^2x_{2,1}x_{3,2}$
%(x_{1123,1222})^{1234,1234} - (x_{1123,1222})^{1234,2314}$
as
\begin{equation*}
\begin{gathered}
    \Big( \ntnsp(x_{1123,1222})^{1234,1234} - (x_{1123,1222})^{1234,2134}\Big)
\,+\,
\Big(\ntnsp(x_{1123,1222})^{1234,2134} - (x_{1123,1222})^{1234,2314}\Big)\\
= \det \ntnsp\begin{bmatrix} x_{1,1} & x_{1,2} \\ x_{1,1} & x_{2,2} \end{bmatrix}\ntnsp x_{2,2} x_{3,2} 
\,+\,
x_{1,2}\det \ntnsp \begin{bmatrix} x_{1,1} & x_{1,2} \\ x_{2,1} & x_{2,2} \end{bmatrix}\ntnsp x_{3,2}, 
\end{gathered}
\end{equation*}
and to write 
 $x_{\smash{1,2}}^2x_{2,1}x_{3,2}
  %, \quad
  %x_{1,1}x_{1,2}x_{2,2}x_{3,2} -
  -
  \smash{x_{1,2}^2}x_{2,2}x_{3,1}$
as
\begin{equation*}
\Big(\ntnsp(x_{1123,1222})^{1234,2314} - (x_{1123,1222})^{1234,2341}\Big)
= x_{1,2}^2 \det \ntnsp\begin{bmatrix} x_{2,1} & x_{2,2} \\ x_{3,1} & x_{3,2} \end{bmatrix}\ntnsp.
\end{equation*}

\section{Main results}\label{s:main}
%\section{Criteria for ratios of principal submatrices}\label{s:main}

Let $\tp n$ be the set of totally positive $n \times n$ matrices.
To characterize ratios of products of permanents which are bounded above and/or nontrivially bounded below on the set $\tp n$, we first consider necessary conditions on the multisets of rows and columns appearing in such ratios.
%Consider a ratio of the form
Let
\begin{equation}\label{eq:mainratio}
    R(x) = \frac{\perm(x_{I_1,I'_1})\perm(x_{I_2,I'_2})\cdots\perm(x_{I_r,I'_r})}{\perm(x_{J_1,J'_1})\perm(x_{J_2,J'_2})\cdots\perm(x_{J_q,J'_q})},
\end{equation} 
be such a ratio, where 
\begin{equation}\label{eq:multisetseqs}
(I_1, \dotsc, I_r), \quad
(I'_1,\dotsc,I'_r), \quad
(J_1,\dotsc,J_q), \quad
(J'_1,\dotsc,J'_q)
\end{equation}
are multisets of $[n]$ satisfying $|I_k| = |I'_k|$,
$|J_k| = |J'_k|$ for all $k$.
In order for
%t is impossible for 
$R(x)$ to be bounded above or nontrivially bounded below on $\tp n$ the multisets \eqref{eq:multisetseqs} must
be related in terms of
%satisfy a condition
%which can be expressed in terms of 
an operation which we call
%A necessary condition for $R(x)$ to be bounded on the set of $n \times n$ totally positive matrices can be stated in terms of a 
%certain 
{\em multiset union}.
%of the multisets (\ref{eq:multisetseqs}).
Given multisets $M = 1^{\alpha_1} \cdots n^{\alpha_n}$, $O = 1^{\beta_1} \cdots n^{\beta_n}$ of $[n]$, define their multiset union to be
\begin{equation}\label{eq:multiuniondef}
M \multiu O \defeq 1^{\alpha_1 + \beta_1} \cdots n^{\alpha_n + \beta_n}.
\end{equation}
%(Change $\alpha$, $\beta$ above and in previous section to $\gamma$, $\delta$.)
For example, $1124 \multiu 1233 = 11122334$.
%We have the following.

%($\star$ Say something about nontrivial lower bounds.  The following proposition says that we can have an upper bound or a nontrivial lower bound only if the multiset condition is satisfied.)

\begin{prop}\label{p:ST0necc}
Given multiset sequences as in \eqref{eq:multisetseqs},
a ratio 
%$R(x)$ as in 
\eqref{eq:mainratio}   
can be bounded above or nontrivially bounded below on $\tp n$ 
%the set of all totally positive matrices
only if we have the multiset equalities
\begin{equation}\label{eq:ST0}
    I_1 \Cup \cdots \Cup I_r = J_1 \Cup \cdots \Cup J_q,
    \qquad
    I'_1 \Cup \cdots \Cup I'_r = J'_1 \Cup \cdots \Cup J'_q.
\end{equation}
\end{prop}
%\begin{lem} If a ratio $$R=\frac{\text{per}(A_{I_1,I'_1})\text{per}(A_{I_2,I'_2})...\text{per}(A_{I_r,I'_r})}{\text{per}(A_{J_1,J'_1})\text{per}(A_{J_2,J'_2})...\text{per}(A_{J_q,J'_q})}$$
%\end{lem}
%is bounded as a function on totally positive matrices, then $R$ satisfies %ST0 condition.
\begin{proof} Omitted.
%Given a multiset $K$, let $\mu_i(K)$ denote the multiplicity of $i$ in $K$, and define
% \begin{equation}
%     \alpha_i = \sum_{k=1}^r \mu_i(I_k),
%     \quad
%      \beta_i = \sum_{k=1}^r \mu_i(I'_k),
%     \quad
%      \alpha'_i = \sum_{k=1}^q \mu_i(J_k),
%     \quad
%      \beta'_i = \sum_{k=1}^q \mu_i(J'_k).
% \end{equation}
% Assume that the multiset equalities (\ref{eq:ST0}) do not hold, e.g., for
% some $i$ we have $\alpha_i \neq \alpha'_i$. Let $A$ be a totally positive matrix and construct a family of matrices $(A(t))_{t > 0}$ by scaling row $i$ of $A$ by $t$.
% Clearly, each matrix $A(t)$ is totally positive, since each minor $\det(A(t)_{I,J})$ equals either $\det(A_{I,J})$ or $t$ times this.
% Furthermore, we have $R(A(t))=t^{\alpha_i-\alpha'_i}R(A)$, since each permanent with row multiset $K$ containing $i$ is scaled by $t^{\mu_i(K)}$. 
% Thus, 
% by letting $t$ approach $0$ or $+\infty$, we can make $R(A(t))$ arbitrarily large or arbitrarily close to $0$.
% The same is true if we have
% $\beta_i \neq \beta'_i$.
\end{proof}

To state sufficient conditions for the boundedness of
%characterize the 
ratios \eqref{eq:mainratio}
%which are indeed bounded, 
we observe that it is possible to bound
the permanent above and below as follows.
%\subsection{Put this somewhere}
%($\star$ Introduce this.)
\begin{prop}\label{p:perminmax}
     For any $n \times n$ totally nonnegative matrix $A = (a_{i,j})$ we have
     \begin{equation}
        a_{1,1} \cdots a_{n,n} \leq \perm(A) \leq n!\cdot a_{1,1} \cdots a_{n,n}. 
    \end{equation}
\end{prop}
\begin{proof} Omitted.
    % The first inequality follows from the fact that $\permmon aw > 0$ for all $w \in \sn$.  The second inequality follows from the fact (\Cref{t:DGS}) that
    % $a_{1,1} \cdots a_{n,n} \leq \permmon aw$ for all $w \in \sn$.   
\end{proof}
% \subsection{The original material}
% Let $\alpha$ be a sequence of row index sets and $\alpha'$ be a sequence of column index sets in the numerator of a ratio. Similarly, let $\beta$ be a sequence of row index sets and $\beta'$ be a sequence of column index sets in the denominator of a ratio. 

% \begin{defn} [\cite{boocher2008generators}]
% For $i\in\{1,2,\ldots,n\}$, let
% $f_\alpha(i)$ be the number of index sets in $\alpha$ that contain
% $i$.  
% %\begin{defn}[ST0 Condition]
% If
% $f_\alpha(i)=f_\beta(i)$ and $f_{\alpha'}(i)=f_{\beta'}(i)$ for all $i$, we say the ratio satisfies the ST0 condition.
% \end{defn}

% ($\star$ Mention somewhere that ratios are always bounded below by $0$ and that by "bounded" we mean "bounded above".)

% ($\star$ Define multiset union and give an example.)

% In the next lemma we show that ST0 condition is necessary condition for a ratio (\ref{eq:R}) to be bounded.

% The following theorem shows that the ST0 condition is also the sufficient condition for a ratio involving permanents of principal submatrices only.

%\section{Some progress with general case}

%($\star$ Introduce this.)
Now we state our main result, which characterizes ratios $R(x)$ as in \eqref{eq:mainratio} which are bounded above for $x \in \tp n$.
\begin{thm}\label{t:main}
Let rational function
\begin{equation}\label{eq:mainratiox}
    R(x) = \frac{\perm(x_{I_1,I'_1})\perm(x_{I_2,I'_2})\cdots\perm(x_{I_r,I'_r})}{\perm(x_{J_1,J'_1})\perm(x_{J_2,J'_2})\cdots\perm(x_{J_q,J'_q})}
\end{equation}    
have index sets which satisfy \eqref{eq:ST0}, and define
%\begin{equation}\label{eq:mainST0}
%\begin{gathered}
%    M \defeq I_1 \Cup \cdots \Cup I_r = J_1 \Cup \cdots \Cup J_q,
%    \qquad
%    O \defeq I'_1 \Cup \cdots \Cup I'_r = J'_1 \Cup \cdots \Cup J'_q,\\
%    m = |I_1| + \cdots + |I_r| = |J_1| + \cdots + |J_q|.
%    \end{gathered}
%\end{equation}
%Define permutations $v,w,u,y \in \mfs{m}$ by (something like)
%Define 
matrices $C = (c_{i,j})$, $C^* = (c^*_{i,j})$, $D = (d_{i,j})$, $D^* = (d^*_{i,j})$ by
\begin{equation}\label{eq:CandD}
(x_{I_1,I'_1})^{e,e} \cdots (x_{I_r,I'_r})^{e,e} = \prod x_{i,j}^{c_{i,j}},
\qquad
(x_{J_1,J'_1})^{e,e} \cdots (x_{J_q,J'_q})^{e,e} = \prod x_{i,j}^{d_{i,j}},
\end{equation}
%and define matrices $C^*$, $D^*$ as in 
and \eqref{eq:cstar}.
Then $R(x)$ is bounded above
on the set of totally positive matrices
%above 
%and nontrivially bounded below (or just say "bounded"?) 
if and only if 
%we have 
$C^* \leq D^*$
in the componentwise order.
%$w(D)$ and $w(C)$ are comparable 
%in ($\star$ what symmetric group or what Bruhat order on double cosets?).
%$u^{-1}y \leq v^{-1}y$.
%In this case, bounds are
In this case, it is bounded above by
%for all $n \times n$ totally positive matrices $A$ we have
%\begin{equation}%\label{eq:mainupperbound}
%\begin{cases}
%\frac{1}{|J_1|! \cdots |J_q|!} \;\leq\; 
%R(A) \;\leq\; 
$|I_1|! \cdots |I_r|!\;$.
%n   &\text{if $w(D) \leq w(C)$},\\
%    \frac{1}{|I_1|! \cdots |I_q|!} \leq R \leq |J_1|! \cdots |J_r|! 
%    &\text{if $w(C) \leq w(D)$}.
%    \end{cases}
%\end{equation}
\end{thm}
\begin{proof}
Suppose that $C^* \nleq D^*$.
%$w(C)$ and $w(D)$ are incomparable in the Bruhat order (on what?). 
Then for some indices $(k,\ell)$ we have $c^*_{k,\ell} > d^*_{k,\ell}$.
Define the matrix $B(t) = (b_{i,j}(t))$ by
\begin{equation*}
    b_{i,j}(t) = \begin{cases}
        t &\text{if $i \leq k$ and $j \leq \ell$},\\
        1 &\text{otherwise}.
    \end{cases}
\end{equation*}   
Now, we have $R(B(t))=\frac{p(t)}{q(t)}$ where $\deg (p(t))=c^*_{i,j}>d^*_{i,j}=\deg (q(t))$.
Thus we have \begin{equation*}
    \lim_{t \rightarrow \infty} R(B(t)) = t^{c^*_{i,j}-d^*_{i,j}}= \infty.
    \end{equation*}

% ($\star$ Now explain what $R(B(t))$ is, and why we have
% \begin{equation*}
%     \lim_{t \rightarrow \infty} R(B(t)) = \infty.)
%     \end{equation*}

Assume therefore that we have $C^* \leq D^*$ and let $A$ be any $n \times n$ totally positive matrix.
Applying the inequalities of 
\Cref{p:perminmax} to the numerator and denominator of $R(A)$
respectively, we see that $R(A)$ is at most
\begin{equation}
%\begin{aligned}
%    R &\leq
%    \frac{|I_1|!(A_{I_1,I'_1})^{e,e} \cdots |I_r|!(A_{I_r,I'_r})^{e,e}}
%    {\perm(A_{J_1,J'_1}) \cdots \perm(A_{J_q,J'_q})}
%    \leq 
\frac{|I_1|!(A_{I_1,I'_1})^{e,e} \cdots |I_r|!(A_{I_r,I'_r})^{e,e}}{(A_{J_1,J'_1})^{e,e} \cdots (A_{J_q,J'_q})^{e,e}} 
    = \frac{|I_1|! \cdots |I_r|! \prod a_{i,j}^{c_{i,j}}}{\prod a_{i,j}^{d_{i,j}}}.
    \end{equation}
    By \Cref{t:bruhatdiff}, this is at most $|I_1|! \cdots |I_r|!\ $.
    

\end{proof}
Observe that \Cref{t:main} guarantees no nontrivial lower bound for $R(x)$ and gives an upper bound which is sometimes tight.
Indeed the ratio 
%\begin{equation*}
    $\smash{ \frac{x_{1,2} x_{2,1}}{x_{1,1}x_{2,2}}}$
%\end{equation*}
attains all values in the open interval $(0,1)$ as $x$ varies over matrices in $\tp 2$. On the other hand, special cases of the ratios in \Cref{t:main} can be shown to have both upper and nontrivial lower bounds.
\begin{cor}\label{cor:bothbounds}
For ratio $R(x)$ and matrices $C$, $D$ defined as in \Cref{t:main},
if $C = D$, then $R(x)$ is bounded above and below by
\begin{equation}\label{eq:corupperbounds}
    \frac{1}{|J_1|! \cdots |J_q|!} \leq R(x) \leq |I_1|! \cdots |I_r|! \;,
\end{equation}
for $x \in \tp n$.
\end{cor}
% ($\star$ Comment in plain English that for principal submatrices the ST0 condition
% is equivalent to the existence of upper and lower bounds.)
For example, consider the ratio
\begin{equation}
    R(x) = \frac{\perm(x_{12,34})\perm(x_{34,12})}
    {x_{1,3}x_{2,4}x_{3,1}x_{4,2}}
\end{equation}
with
$|I_1| = |I_2| = 2$, $|J_1| = |J_2| = |J_3| = |J_4| = 1$, and
\begin{equation}
    C = D = \begin{bmatrix} 
    0 & 0 & 1 & 0 \\ 
    0 & 0 & 0 & 1 \\
    1 & 0 & 0 & 0 \\
    0 & 1 & 0 & 0
    \end{bmatrix}.
\end{equation}
By \Cref{cor:bothbounds}, we have
%\begin{equation*}
%    \frac{1}{1!^4} \leq R(x) \leq 2!^2.
%    \end{equation*}
$1 \leq R(x) \leq 4$.
    It is easy to see that $R(x)$ attains values arbitrarily close to $4$ as $x$ approaches the matrix of all ones.
    %the upper bound is tight.  
%    ($\star$ What about the lower bound?)
It is also possible to show that $R(x)$
%$ = 1 - 2\epsilon + \epsilon^2$ 
attains values arbitrarily close to $1$.  Indeed, consider the matrix $A = A(\epsilon) = (a_{i,j})$ defined by
\begin{equation}
    A(\epsilon) = \begin{bmatrix}
        1 & 1 & \epsilon & \epsilon^3 \\
    1 & 2 & 1 & \epsilon \\
    \epsilon & 1 & 2 & 1 \\
    \epsilon^3 & \epsilon & 1 & 1
    \end{bmatrix},
\end{equation}%\begin{equation*}
%\end{equation*}
where $\epsilon$ is positive and close to $0$.
To see that $A(\epsilon)$ is totally positive, 
%for small $\epsilon > 0$, 
it suffices to verify the positivity of the
sixteen minors $\det(A_{[a_1,b_1],[a_2,b_2]})$ indexed by pairs of 
%p sets are 
intervals, at least one of which contains $1$~\cite[Theorem\,9]{FominTPTest}.
Observe that we have
$a_{1,j} > 0$ and $a_{i,1} > 0$ for all $i,j$.  Also,
% \begin{equation*}
% \begin{gathered}
%     \det(A_{12,12}) = 1,\qquad
%     \det(A_{12,23}) = \det(A_{23,12}) = 1 - 2\epsilon,\qquad
%     \det(A_{12,34}) = \det(A_{34,12}) = \epsilon^2 - \epsilon^3,\\
%     \det(A_{123,123}) = 1 + 2 \epsilon - 2 \epsilon^2,\qquad
%     \det(A_{123,234}) = \det(A_{234,123}) = 1 - 4\epsilon + \epsilon^2 + 3\epsilon^3,\\
%     \det(A) = 
% 4\epsilon-6\epsilon^2-2\epsilon^3+9\epsilon^4-2\epsilon^5-3\epsilon^6.    
%     \end{gathered}
% \end{equation*}
\begin{equation*}
\begin{aligned}
    &\det(A_{12,12}) = 1,\\
    &\det(A_{12,23}) = \det(A_{23,12}) = 1 - 2\epsilon,\\
    &\det(A_{12,34}) = \det(A_{34,12}) = \epsilon^2 - \epsilon^3,\\
    &\det(A_{123,123}) = 1 + 2 \epsilon - 2 \epsilon^2,\\
    &\det(A_{123,234}) = \det(A_{234,123}) = 1 - 4\epsilon + \epsilon^2 + 3\epsilon^3,\\
    &\det(A) = 
4\epsilon-6\epsilon^2-2\epsilon^3+9\epsilon^4-2\epsilon^5-3\epsilon^6.    
    \end{aligned}
\end{equation*}
It follows that 
%Now observe that 
we have
\begin{equation*}
    \lim_{\epsilon \rightarrow 0^+} R(A) = 
    \lim_{\epsilon \rightarrow 0^+}
    \frac{(\epsilon^2 - \epsilon^3)^2}{\epsilon^4} = 
\lim_{\epsilon \rightarrow 0^+} 1 - 2\epsilon + \epsilon^2 = 1.
    \end{equation*}
    
%\end{equation*}
%$\epsilon$ approaches %$0^{+}$,

In the case that all submatrices in \eqref{eq:mainratiox} are principal, the necessary condition \eqref{eq:ST0} for boundedness is in fact sufficient to guarantee the existence of upper and nontrivial lower bounds.
\begin{cor}\label{cor:principalbounds}
For ratio $R(x)$ as in \Cref{t:main},
if only principal submatrices appear
%in $R(x)$ are principal, 
($I_k = I'_k$, $J_k = J'_k$ for all $k$), then
$R(x)$ is bounded above and below as in \eqref{eq:corupperbounds}.
%\begin{equation*}
%    \frac{1}{|J_1|! \cdots |J_q|!} \leq R \leq |I_1|! \cdots |I_r|! \;.
%\end{equation*}
\end{cor}
\begin{proof} Omitted.
%    When all submatrices in the ratio $R$ are  
%(Explain why?)   
% For principal submatrices $x_{I_1,I_1},\dotsc$,
%     the condition (\ref{eq:ST0}) implies the equality of the matrices $C$ and $D$ in (\ref{eq:CandD}):
%     this matrix is diagonal with $(i,i)$ entry equal to the multiplicity of $i$ in $I_1 \multiu \cdots \multiu I_r$.
\end{proof}

For example, consider the ratio 
%(\ref{eq:kotelratio})
\begin{equation}
%\frac1{|I \cup J|! ~| I \cap J|!}
%\leq
\frac{\perm(x_{I,I})\;\perm(x_{J,J})}
{\perm(x_{I\cup J, I\cup J})\;\perm(x_{I\cap J, I\cap J})}
%\leq
%|I|!~|J|!
%\end{equation}
\end{equation}
coming from the (false) permanental version \eqref{eq:kotelperm} of Koteljanskii's inequality \eqref{eq:koteljanskii}.
By \Cref{cor:principalbounds}, the four principal submatrices of $x$ imply that the exponent matrices $C$ and $D$ are equal and diagonal with $(i,i)$ entry equal to the multiplicity of $i$ in $I \multiu J$. Thus \Cref{cor:bothbounds} gives the lower and upper bounds
\begin{equation}
    \frac 1{|I\cup J|!\; |I \cap J |!}, \qquad
    |I|! \; |J|!
\end{equation}
as claimed in \eqref{eq:kotelratio}.  These bounds are not in general tight. Consider the special case
\begin{equation}\label{eq:worsebounds}
\frac{1}{3!1!}\leq
\frac{\perm(x_{12,12})\perm(x_{23,23})}{\perm(x_{123,123})\perm(x_{2,2})}
\leq (2!)^2, \qquad
%\end{equation}
%with 
%\begin{equation}
    C = D = \begin{bmatrix}
        1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 
    \end{bmatrix}.
\end{equation}
%multiset unions (\ref{eq:ST0}) equal to %$1223$. 
%We improve (\ref{eq:worsebounds}) as follows.
A more careful analysis leads to 
%an 
improved bounds of $1/2$ and $2$,
%an improved 
%By \Cref{cor:bothbounds} we have
%If the index sets in (\ref{eq:mainratio}) do not satisfy (\ref{eq:mainST0}).  Then by \Cref{p:unbounded}, the ratio is
%not bounded above or below.
%Suppose therefore that the index sets do satisfy (\ref{eq:mainST0}).
%    $\star$ Do this.
%\end{proof}
%($\star$ Mention that the stated bounds are not tight. Use the fact that every TNN matrices is the limit of a sequence of TNN matrices.)
%($\star$  Do some examples with bounds guaranteed by the theorem,
%and with best possible bounds computed by other arguments such as
%the subtraction-free property.)
%\begin{prop}
%For $x \in \tp 3$ we have
%\begin{equation}\label{eq:half2}\frac{1}{2}\leq\frac{\perm(x_{12,12})\perm(x_{23,23})}{\perm(x_{123,123})\perm(x_{2,2})}\leq 2,
%\end{equation}
%\end{prop}
%\begin{proof}
 since the difference
 %with the first inequality followings from expressing 
$2\;\ntnsp\perm(x_{12,12})\perm(x_{23,23})-\perm(x_{123,123})\perm(x_{2,2})$ equals
 %and grouping terms as
\begin{equation*}
%    \begin{aligned}
%    &(x_{11}x^2_{22}x_{33}-x_{13}x^2_{22}x_{31}) +(x_{12}x_{21}x_{22}x_{33}-x_{12}x_{22}x_{23}x_{31})
%    + (x_{12}x_{21}x_{23}x_{32}-x_{13}x_{21}x_{22}x_{32})\\
%    &\quad + x_{11}x_{22}x_{23}x_{32}+x_{12}x_{21}x_{23}x_{32}\\
%&= 
\det(x_{13,13})x^2_{22}+\det(x_{23,13})x_{12}x_{22}+\det(x_{12,23})x_{21}x_{32}+x_{11}x_{22}x_{23}x_{32}+x_{12}x_{21}x_{23}x_{32},
%\end{aligned}
\end{equation*}
%Similarly, the second inequality follows from expanding
and the difference $2\;\ntnsp\perm(x_{123,123})\perm(x_{2,2})-\perm(x_{12,12})\perm(x_{23,23})$ equals
%and grouping terms as
\begin{equation*}
%    \begin{aligned}
%2 \perm(x_{123,123})\perm(x_{2,2}) - \perm(x_{12,12})\perm(x_{23,23})\\
%&=
x_{11}x^2_{22}x_{33}+\det(x_{23,23})x_{12}x_{21}+2x_{12}x_{22}x_{23}x_{31}+x_{11}x_{22}x_{23}x_{32}+2x_{13}x_{21}x_{22}x_{32}+2x_{13}x^2_{22}x_{31}.
%\end{aligned}
\end{equation*}
%\end{proof}
%According to Corollary~(\ref{cor:bothbounds}) a ratio below %$\frac{\perm(A_{12|12})\perm(A_{13|13})}{\perm(A_{123|123})\perm(A_{2|2})}$ 
%has the following lower and upper bounds
%This bounds can be improved. The following two inequalities 
%suggest tighter lower and upper bounds 
%$\star$ Can we say whether the bounds above are tight?
%Furthermore, t
The authors believe that even these bounds are not tight.  The smallest and greatest values we have found for the ratio in \eqref{eq:worsebounds}
%(\ref{eq:half2}) that we have found 
are $2/3$ and $121/114$, respectively.
%\end{obs}

% \section{Future directions}\label{s:open}
 It would be interesting to characterize the ratios \eqref{eq:R} which are bounded by $1$, or to find
 examples of such ratios.
 %i.e., to solve the following problem.
 \bp 
 Characterize the differences
%\begin{equation}\label{eq:perprodminusperprod}
 $\perm(x_{J_1,J'_1}) \cdots \perm(x_{J_q,J'_q}) - 
     \perm(x_{I_1,I'_1}) \cdots \perm(x_{I_r,I'_r})$
% \end{equation}
 which are totally nonnegative polynomials.
\ep
% %By Proposition It seems that boundaries in the corollary could be significantly improved. 
% %For example, 
% To consider a special case, i
%For instance, it is possible to show that f
%As a first step, one could provide examples of such
%polynomials.
\bp 
%In particular, s
Show that
%For small $n$, the sets 
%$I = [2n] \ssm 2\mathbb Z$, $J = [2n] \cap 2 \mathbb Z$, 
the 
%define a totally nonnegative
polynomial
%\begin{equation}\label{eq:sihong}
$\perm(x_{[n],[n]}) \perm(x_{[n+1,2n],[n+1,2n]}) - 
\perm(x_{I,I}) \perm(x_{J,J})$
%\end{equation}
is totally nonnegative
when $I = [2n] \ssm 2\mathbb Z$, $J = [2n] \cap 2 \mathbb Z$.
\ep
% If this polynomial is totally nonnegative in general, then
% it provides a permanental analog of the known totally nonnegative polynomial
% \begin{equation*}
%     \det(x_{I,I}) \det(x_{J,J}) - 
%     \det(x_{[n,n]})
%     \det(x_{[n+1,2n],[n+1,2n]}).
% \end{equation*}
% %
% %\perm(x_{\{1,2\},\{1,2\}})
% %\perm(x_{\{3,4\},\{3,4\}}) - 
% %    \perm(x_{\{1,3\},\{1,3\}}) \perm(x_{\{2,4\},\{2,4\}})
% %\end{equation*}
% %the following inequalities hold for small values of $n$.   
% %Thus it is natural to conjecture the following.
% %More generally, we have the following.
% \begin{conj}
% %Show that 
% The polynomial (\ref{eq:sihong}) is totally nonnegative for all 
% %positive 
% $n$.
% %For any $2n \times 2n$ totally positive matrix $A$:
% %$$    
% %\perm(A_{[1,3,5...2n-1],[1,3,5...2n-1]})\perm(A_{[2,4,6...2n],[2,4,6...2n]}) \leq $$ 
% %$$\leq %\perm(A_{[1,2,3...n],[1,2,3...n]})\perm(A_{[n+1,n+2,n+3...2n],[n+1,n+2,n+3...2n]})
% %$$
% \end{conj}
% %The inequality above for $4 \times 4$ matrix $A$ is `subtraction-free' in terms of positive weights of the planar network parameterization (see \cite{fomin2000total} for definitions).  
% Other families of possible inequalities are suggested by the known inequalities appearing in
% %Some experimentation suggests that we can extend the second inequality in 
% \Cref{p:perminmax}.
% In particular,
% the first inequality there compares the permanent to a product of permanents of $1 \times 1$ matrices.  Comparing further to products of permanents of the form
% \begin{equation}\label{eq:permprodgen}
% \perm(x_{I_1,I_1}) \cdots \perm(x_{I_r,I_r}),
% \end{equation}
% we obtain polynomials
% of the forms
% \begin{equation}
%     \perm(x) - \perm(x_{I_1,I_1}) \cdots \perm(x_{I_r,I_r}),
%     \qquad
%     \perm(x_{I_1,I_1}) \cdots \perm(x_{I_r,I_r}) - x_{1,1} \cdots x_{n,n},
% \end{equation}
% which are totally nonnegative because they belong to $\spn_{\mathbb N}\{ x^{e,w}
% %x_{1,w_1} \cdots x_{n,w_n} 
% \,|\, w \in \sn \}$.
% It is natural to ask if the cardinalities of the index sets determine whether a difference of the form (\ref{eq:perprodminusperprod}) is totally nonnegative, but this is not the case.
% %cardinalities alone do not answer this question.  
% It is natural then to ask how {\em averages} of such products compare to one another.  This problem is open.  (See \cite{BJMajor}, \cite[Prob.\,5.3]{SkanSoskinBJArx}.)
% \bp Characterize the pairs of partitions $\lambda = (\lambda_1,\dotsc,\lambda_r)$,
% $\mu = (\mu_1,\dotsc,\mu_q)$ such that
% \begin{equation}\label{eq:bjperm}
% \sumsb{(I_1,\dotsc,I_r)\\|I_k| = \lambda_k}
% \frac{\perm(x_{I_1,I_1}) \cdots \perm(x_{I_r,I_r})}{\tbinom{n}{\lambda_1,\dotsc,\lambda_r}}
% -
% \sumsb{(J_1,\dotsc,J_q)\\|J_k| = \mu_k}
% \frac{\perm(x_{J_1,J_1}) \cdots \perm(x_{J_q,J_q})}{\tbinom{n}{\mu_1,\dotsc,\mu_r}}
% \end{equation}
% is totally nonnegative.
% \ep
% % as follows.

% Now consider generalizing the second inequality in \Cref{p:perminmax} to products of permanents of the form (\ref{eq:permprodgen}). Differences of the form
% \begin{equation}\label{eq:nottnn}
% \frac{\perm(x_{I_1,I_1}) \cdots \perm(x_{I_r,I_r})}{|I_1|! \cdots |I_r|!} - 
% \frac{\perm(x)}{n!}
% \end{equation}
% are {\em not} totally nonnegative, while differences of the form
% \begin{equation}
% x_{1,1} \cdots x_{n,n} - \frac{\perm(x_{I_1,I_1}) \cdots \perm(x_{I_r,I_r})}{|I_1|! \cdots |I_r|!}
% \end{equation}
% are (by \Cref{p:perminmax}).  It is natural then to ask about the averages of differences (\ref{eq:nottnn}), over all set partitions $(I_1,\dotsc,I_r)$ of a partition $\lambda$.

%\newpage
%\bp\label{p:avgminusperm}
%Decide if for fixed $\lambda = (\lambda_1,\dotsc,%\lambda_r)$, the polynomial
%\begin{equation*}
%     \sumsb{(I_1,\dotsc,I_r)\\|I_k| = \lambda_k} \perm(x_{I_1,I_1}) \cdots \perm(x_{I_r,I_r})
%     -
%     \perm(x)
% \end{equation*}
% is totally nonnegative.
%\ep

% %Show that for any $n \times n$ totally positive matrix $A$, we have
% %$$\frac{\perm(A)}{n!} \leq \frac{\perm(A_{I_1,I_1})\perm(A_{I_2,I_2})...\perm(A_{I_r,I_r})}{ |I_1|! \cdot |I_2|! \cdot...\cdot |I_r|!}$$ 


% %\begin{ex}
% %($\star$ Extend this to average.)
% To illustrate (\ref{eq:nottnn}) and Problem~\ref{p:avgminusperm}, let us consider the case that $n = 3$. It is straightforward to show that
% \begin{equation}\label{eq:121233}
% \frac{\perm(x_{12,12}) x_{3,3}}{2!1!} - \frac{\perm(x)}{3!},
% \end{equation}
% equivalently, $3\perm(x_{12,12})x_{3,3} - \perm(x)$, is totally nonnegative because the latter expression equals a sum of matrix minors.
% Similarly,
% \begin{equation}\label{eq:232311}
% \frac{\perm(x_{23,23}) x_{1,1}}{2!1!} - \frac{\perm(x)}{3!},
% \end{equation}
% is totally nonnegative.
% On the other hand,
% \begin{equation}\label{eq:131322}
% \frac{\perm(x_{13,13}) x_{2,2}}{2!1!} - \frac{\perm(x)}{3!},
% \end{equation}
% is not, because its evaluation at
% %when
% %we evaluate $x$ at
% \begin{equation*}
%  \begin{bmatrix}
%         1 & 1 & 0 \\
%         1 & 1 & 1 \\
%         1 & 1 & 1 
%     \end{bmatrix}
% \end{equation*}
% is negative.
% On the other hand, two times
% the sum of the three differences (\ref{eq:121233}) -- (\ref{eq:131322}) is
% \begin{equation*}
%  2x_{1,1}x_{2,2}x_{3,3} - x_{1,2}x_{2,3}x_{3,1} - x_{1,3}x_{2,1}x_{3,2},
%  \end{equation*}
%  which is totally nonnegative by \Cref{t:DGS}.

%Let $A$ be a $3 \times 3$ totally positive matrix . Then $$\frac{\text{per}(A)}{3!} \leq \frac{\text{per}(A_{12,12})a_{3,3}}{2!\cdot 1!}$$ 
%Indeed, $$ \text{per}(A) \leq 3a_{11}a_{22}a_{33}+3a_{21}a_{12}a_{33}$$
%$$a_{11}a_{22}a_{33} + a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} + a_{12}a_{21}a_{33} + 
%a_{13}a_{21}a_{32} + a_{13}a_{22}a_{31} \leq 3a_{11}a_{22}a_{33}+3a_{21}a_{12}a_{33}$$
%We can split all the terms in pairs:
%$$0 \leq (a_{11}a_{22}a_{33}-a_{11}a_{22}a_{33})+(a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32})+(a_{11}a_{22}a_{33}-a_{13}a_{22}a_{31})+$$
%$$+(a_{21}a_{12}a_{33}-a_{12}a_{23}a_{31})+
%(a_{21}a_{12}a_{33}-a_{12}a_{21}a_{33})+
%(a_{21}a_{12}a_{33}-a_{13}a_{22}a_{31})$$
%\end{ex}
%All of the six pairs above are non-negative because of  the theorem \cite[Theorem~2]{drake2004two}.




%\acknowledgements{\lipsum[20]}

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