\documentclass[11pt]{amsart} \pdfoutput=1 \usepackage[usenames,dvipsnames]{xcolor} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[letterpaper,margin=1in]{geometry} \usepackage{lmodern} \usepackage{microtype} \usepackage{etoolbox} \usepackage{mathtools} \usepackage{amssymb} \usepackage{enumitem} \usepackage[round]{natbib} \usepackage{hyperref} \numberwithin{equation}{section} \patchcmd{\section}{\scshape}{\fontfamily{ppl}\bfseries\scshape\selectfont}{}{} \hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=red!65!black} \setlist{nosep,leftmargin=*} \newtheorem{theorem}{Theorem}[section] \newtheorem{prop}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{rmk}[theorem]{Remark} \theoremstyle{definition} \newtheorem{defn}[theorem]{Definition} \newcommand{\reals}{\mathbb{R}} \newcommand{\E}{\mathbb{E}} \newcommand{\nlsum}{\sum\nolimits} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\per}{per} \DeclareMathAlphabet{\mathbbold}{U}{bbold}{m}{n} \newcommand{\ii}{\mathbbold{i}} % \DeclareMathOperator{\Cov}{Cov} \DeclareMathOperator{\Var}{Var} \newcommand{\one}{\mathbf 1} \newcommand{\cL}{\mathcal L} \newcommand{\cE}{\mathcal E} \newcommand{\cZ}{\mathcal Z} \newcommand{\T}{\mathbb T} \newcommand{\dotast}{\mathbin{\dot\ast}} \hypersetup{ pdftitle={A Koteljanskii inequality for permanents}, pdfauthor={Suvrit Sra} } \setlist[itemize]{topsep=4pt,itemsep=2pt,parsep=2pt} \setlist[enumerate]{topsep=4pt,itemsep=2pt,parsep=2pt} \makeatletter \newcommand{\arxivlink}[1]{% \thanks{Also on arXiv: \url{https://arxiv.org/abs/#1}.}} \makeatother \begin{document} \title{A Koteljanskii inequality for permanents} \author[Suvrit Sra]{Suvrit Sra\textsuperscript{\normalfont*}} \thanks{*\,TU Munich, Dept.~of Mathematics, School of CIT; Garching, Germany. Email: \texttt{s.sra@tum.de}.} \begin{abstract} We prove a permanental analogue of Koteljanskii's inequality. If \(A\) is an inverse \(M\)-matrix that becomes symmetric after a positive diagonal similarity, then \(\per(A_{S\cup T})\per(A_{S\cap T})\ge\per(A_S)\per(A_T)\) for all \(S,T\subseteq[n]\), where \(A_S\) is the principal submatrix indexed by \(S\). The proof expresses permanents as moments of a complex Gaussian vector and uses Ginibre's correlation inequality. Finally, an explicit counterexample shows that symmetry cannot be dropped. % \end{abstract} \arxivlink{2609.39979} \maketitle \vspace*{-5pt} \section{Introduction}\label{sec:intro} For an \(n\times n\) matrix \(A\) and \(S\subseteq[n]\), write \(A_S=A[S,S]\) for the principal submatrix with row and column set \(S\), and set \(\det(A_\varnothing)=\per(A_\varnothing)=1\). Koteljanskii's inequality \citep{koteljanskii1950theory,koteljanskii1953property} \[ \det(A_{S\cup T})\det(A_{S\cap T})\le\det(A_S)\det(A_T), \qquad S,T\subseteq[n], \] holds for positive semidefinite matrices, and also for \(M\)-matrices and inverse \(M\)-matrices \citep{fallat1998characterization}. Its case \(S\cap T=\varnothing\) is Fischer's inequality, and iterating Fischer's inequality gives Hadamard's inequality \(\det(A)\le\prod_ia_{ii}\). For positive semidefinite matrices the permanent reverses Hadamard's inequality, \(\per(A)\ge\prod_ia_{ii}\) \citep{marcus1963permanent}, and Lieb's permanental inequality \citep{lieb1966} reverses Fischer's: \(\per(A_{S\cup T})\ge\per(A_S)\per(A_T)\) for disjoint \(S,T\). These two reversals suggest the permanental analogue of Koteljanskii's inequality, \begin{equation}\label{eq:lsm} \per(A_{S\cup T})\per(A_{S\cap T})\ge \per(A_S)\per(A_T), \qquad S,T\subseteq[n], \end{equation} where as usual the permanent of an $n\times n$ matrix is defined as \begin{equation} \label{eq:1} \per(A)=\sum_{\pi\in \mathrm{Sym}_n}\prod\nolimits_{i\in [n]}A_{i,\pi(i)}. \end{equation} \vskip4pt \noindent\emph{Which matrices satisfy~\eqref{eq:lsm}}? % Contrary to the determinant, here positive semidefiniteness does not suffice, even with nonnegative entries: the positive semidefinite matrix \(A\) with rows \((1,1,0)\), \((1,2,1)\), \((0,1,1)\) has \(\per(A)\per(A_{\{2\}})=8<9=\per(A_{\{1,2\}})\per(A_{\{2,3\}})\). This obstruction led us to consider inverse \(M\)-matrices instead, as they also satisfy Koteljanskii's inequality. Numerous initial experiments in early 2015 convinced us that for inverse $M$-matrices the inequality~\eqref{eq:lsm} could hold. % \vskip5pt A matrix \(A\) is an inverse \(M\)-matrix if \(A\ge0\) entrywise, \(A\) is invertible, and \(A^{-1}\) has nonpositive off-diagonal entries. We prove~\eqref{eq:lsm} for inverse \(M\)-matrices that are symmetric after a positive diagonal similarity, which we call \emph{symmetrizable}, and show that~\eqref{eq:lsm} can fail without this hypothesis. \begin{theorem}[Koteljanskii for $\per$]\label{thm:main} Let \(A\) be a symmetrizable inverse \(M\)-matrix. Then the set function \(S\mapsto\per(A_S)\) is log-supermodular, i.e., inequality~\eqref{eq:lsm} holds for all \(S,T\subseteq[n]\). \end{theorem} \begin{theorem}[Counterexample]\label{thm:counterexample} The matrix \begin{small} \[ A=\begin{pmatrix} 13&1&7&7&7\\ 1&13&7&7&7\\ 1&1&7&7&7\\ 1&1&1&7&7\\ 1&1&1&1&7 \end{pmatrix} \] \end{small} is a non-symmetrizable inverse \(M\)-matrix, and \(\per(A)\per(A_{\{3,4,5\}})<\per(A_{\{1,3,4,5\}})\per(A_{\{2,3,4,5\}})\). \end{theorem} For the proof of Theorem~\ref{thm:main} it suffices to consider symmetric (and hence, also positive definite) inverse $M$-matrices since diagonal similarity does not change principal permanents. Our starting point is the classical representation of \(\per(A_S)\) as the moment \(\E\prod_{i\in S}|\zeta_i|^2\) of a complex Gaussian vector. % Then, Ginibre's inequality \citep{ginibre1970general} helps us show that the density of \(X=(|\zeta_1|^2,\dots,|\zeta_n|^2)\) % is MTP\(_2\) (multivariate totally positive of order two)---see Proposition~\ref{prop:radial}. This property provides a key building block for a tilting based argument that closes the proof. % Our results leave open which non-symmetrizable inverse \(M\)-matrices satisfy~\eqref{eq:lsm}; Section~\ref{sec:open} lists this and other questions. \subsection{Related work.} \citet{johnson1982inverse} surveys inverse \(M\)-matrices, with a sequel by \citet{johnson2011inverse}; for permanental inequalities on totally positive matrices see \citet{skandera2025permanental}. The MTP property and the FKG inequality for densities are due to \citet{karlin1980classes}. \citet{eisenbaum2014characterization} characterized positively correlated squared Gaussian vectors and extended the characterization to the permanental vectors of Section~\ref{sec:alpha}; see the erratum \citep{eisenbaum2022erratum} and \citet{finner2024positive}, who show that positively associated squared Gaussian vectors need not be MTP\(_2\) and extend this to multivariate $\chi$-square distributions. Positive association alone does not give~\eqref{eq:lsm}, which needs association after the tilt by \(\prod_{i\in S\cap T}X_i\) (Remark~\ref{rmk:assoc}); the example we provide therein also contradicts Corollary~11 of the technical report of \citet{lovasz2003}. \subsection{Notation and definitions}\label{sec:prelim} Throughout, we use the shorthand \([n]=\{1,2,\dots,n\}\). The central class of matrices for this paper is that of inverse $M$ matrices; we define these below. \begin{defn}[\(M\)-matrices] A square matrix \(M=(m_{ij})\) is a nonsingular \emph{\(M\)-matrix} if: (i) \(m_{ij}\le0\) for all \(i\ne j\); and (ii) \(M\) is invertible with \(M^{-1}\ge0\) entrywise. A matrix $A$ is an \emph{inverse $M$-matrix} if \(A=M^{-1}\) for some nonsingular \(M\)-matrix \(M\). % \end{defn} \noindent Let us also formally define log-supermodularity. \begin{defn}[Log-supermodularity] A set function \(f:2^{[n]}\to\reals_{\ge0}\) is \emph{log-supermodular} if \[ f(S\cup T)f(S\cap T)\ge f(S)f(T),\qquad S,T\subseteq[n]. \] \end{defn} We are now ready to prove our main result, Theorem~\ref{thm:main}. \section{Proof of Theorem~\ref{thm:main}}\label{sec:proof} The proof has four steps: Step~1 reduces to symmetric \(A\); Step~2 writes \(\per(A_S)\) as a squarefree moment of \(X=(|\zeta_1|^2,\dots,|\zeta_n|^2)\), where \(\zeta\) is a complex Gaussian vector with covariance \(A\); Step~3 shows that \(X\) has an MTP\(_2\) density, and once the phases are integrated out, this reduces to a correlation inequality of Ginibre; Step~4 finishes by observing that squarefree moments of an MTP\(_2\) density are log-supermodular. \subsection{Step 1, symmetry} Let \(\Delta\) be a positive diagonal matrix such that \(A'=\Delta A\Delta^{-1}\) is symmetric. This is again an inverse \(M\)-matrix: \(A'\ge0\), and \((A')^{-1}=\Delta M\Delta^{-1}\) has the sign pattern of \(M=A^{-1}\). Moreover, for every \(S\subseteq[n]\) and every permutation \(\pi\) of \(S\), $\prod_{i\in S}A'_{i,\pi(i)} =\prod_{i\in S}\frac{\Delta_{ii}}{\Delta_{\pi(i),\pi(i)}}A_{i,\pi(i)} =\prod_{i\in S}A_{i,\pi(i)}$. Thus, \(\per(A'_S)=\per(A_S)\); so henceforth we may assume that \(A\) is symmetric. \subsection{Step 2, Gaussian moments} Now that $A$ is symmetric, we also need it to be positive definite so that we can use it as the covariance matrix of a complex Gaussian vector. Fortunately, a symmetric inverse \(M\)-matrix is automatically positive definite. Indeed, put \(x=A\one\); then \(x>0\), since no row of the invertible nonnegative matrix \(A\) vanishes. Since \(Mx=\one\), the symmetric matrix \(\diag(x)M\diag(x)\) has nonpositive off-diagonal entries and \(i\)-th row sum \(x_i>0\). It is therefore strictly diagonally dominant with positive diagonal, hence positive definite, and so are \(M\) and \(A\). Let \(\zeta\in\mathbb C^n\) be a complex Gaussian vector with covariance \(A\) and density \(\pi^{-n}\det M\,e^{-z^*Mz}\). Let \(X=(|\zeta_1|^2,\ldots,|\zeta_n|^2)\). By the moment theorem for complex Gaussian vectors \citep{reed1962moment}, \(\E\prod_{i\in S}X_i=\E\prod_{i\in S}\zeta_i\overline{\zeta_i}=\per(A_S)\); in particular, all squarefree moments of \(X\) are finite. \subsection{Step 3, MTP\(_2\) density} Next, we wish to show that the density of $X$ is MTP${}_2$. Recall that for a positive density \(p\) on \((0,\infty)^n\), the MTP\(_2\) property is \begin{equation} \label{eq:2} p(x\vee y)p(x\wedge y)\geq p(x)p(y). \end{equation} For $C^2$ densities~\eqref{eq:2} is equivalent to non-negative Hessian off-diagonals, i.e., $\partial_{ij}\log p\geq0$ for \(i\ne j\). We prove this non-negativity for the density of $X$. To that end, first write \(M=D-C\), where \(D=\diag(d_1,\dots,d_n)\) with \(d_i=M_{ii}>0\), and \(C\ge0\) is symmetric with zero diagonal~\citep[Chapter~6]{berman1994}. Let \(\cE\) be the pairs \(\{i,j\}\) with \(C_{ij}>0\); using polar coordinates \(z_i=\sqrt{x_i}e^{\ii\theta_i}\), we observe that \[ z^*Mz=\nlsum_id_ix_i-\nlsum_{\{i,j\}\in\cE}2C_{ij}\sqrt{x_ix_j}\cos(\theta_i-\theta_j). \] Now integrate out \(\theta\) to obtain the density \begin{equation}\label{eq:density} p(x)=\det M\exp\!\left(-\nlsum_i d_ix_i\right)\cZ(J(x)), \qquad J_{\{i,j\}}(x)=2C_{ij}\sqrt{x_ix_j}, \end{equation} where \(\cZ\) is the partition function on $\cE$ defined as follows: For \(e=\{i,j\}\in\cE\) set \(\omega_e(\theta)=\cos(\theta_i-\theta_j)\), where \(\theta\in\T^n=(\reals/2\pi\mathbb Z)^n\). For nonnegative couplings \(J_e\), define \[ \cZ(J) :=\int_{\T^n}\exp\!\left(\nlsum_{e\in\cE}J_e\omega_e(\theta)\right)d\theta, \qquad d\mu_J(\theta)=\cZ(J)^{-1} \exp\!\left(\nlsum_{e\in\cE}J_e\omega_e(\theta)\right)d\theta, \] with normalized Haar measure \(d\theta\). Write \(\E_J\) and \(\Cov_J\) for expectation and covariance under \(\mu_J\); differentiation under the integral then gives \[ \frac{\partial\log\cZ}{\partial J_e}=\E_J[\omega_e], \qquad \frac{\partial^2\log\cZ}{\partial J_e\partial J_f} =\Cov_J(\omega_e,\omega_f). \] The sign of \(\partial_{ij}\log p\) is therefore controlled by the angular means \(\E_J[\omega_e]\) and covariances \(\Cov_J(\omega_e,\omega_f)\). Inequality~\eqref{eq:3} below is a case of Ginibre's inequality \citep{ginibre1970general} for edge observables. We include its proof to make explicit which correlation property is used. \begin{lemma}\label{lem:angular} For every \(e,f\in\cE\), $\E_J[\omega_e]\geq0$, and $\Cov_J(\omega_e,\omega_f)\geq0$. \end{lemma} \begin{proof} The absolutely convergent Fourier expansion $e^{J\cos u} =\sum_{p,q\geq0}\frac{(J/2)^{p+q}}{p!q!}e^{\ii(p-q)u}$ has nonnegative coefficients. The product over edges has the same property, so its integral against \(\cos(\theta_i-\theta_j)\) is nonnegative. Division by \(\cZ(J)>0\) proves the first inequality. For the covariance, introduce two independent angular variables \(\theta,\phi\). With \(H(\theta)=\sum_gJ_g\omega_g(\theta)\), \[ 2\cZ(J)^2\Cov_J(\omega_e,\omega_f) =\int (\omega_e(\theta)-\omega_e(\phi))(\omega_f(\theta)-\omega_f(\phi)) e^{H(\theta)+H(\phi)}\,d\theta\,d\phi. \] The map \((u,v)\mapsto(u+v,u-v)\) is a surjective homomorphism of \(\T^{2n}\), and therefore pushes normalized Haar measure to normalized Haar measure. Orient each edge arbitrarily and write \(u_e=u_i-u_j\) and \(v_e=v_i-v_j\). Under the substitution \(\theta=u+v\), \(\phi=u-v\), \[ \omega_e(\theta)-\omega_e(\phi)=-2\sin u_e\sin v_e, \qquad H(\theta)+H(\phi)=2\sum_gJ_g\cos u_g\cos v_g. \] Expanding the exponential gives \begin{equation} \label{eq:3} \Cov_J(\omega_e,\omega_f) =\frac{2}{\cZ(J)^2} \nlsum_{m\in\mathbb N_0^\cE} \left(\prod\nolimits_g\frac{(2J_g)^{m_g}}{m_g!}\right) \left[ \int_{\T^n}\sin u_e\sin u_f\prod\nolimits_g(\cos u_g)^{m_g}\,du \right]^2\geq0, \end{equation} which completes the proof. \end{proof} We are now ready to formally state the desired MTP$_2$ property. \begin{prop}\label{prop:radial} Let \(A\) be a symmetric inverse \(M\)-matrix, and let \(\zeta\in\mathbb C^n\) % be a complex Gaussian vector with covariance \(A\). Then \(X=(|\zeta_1|^2,\ldots,|\zeta_n|^2)\) has an MTP\(_2\) density on \((0,\infty)^n\). \end{prop} \begin{proof} By~\eqref{eq:density} and the derivatives of \(\log\cZ\) above, for \(i\ne j\), \begin{align*} \partial_i\partial_j\log p(x) &=\nlsum_e\E_J[\omega_e]\,\partial_i\partial_jJ_e(x)+\nlsum_{e,f}\Cov_J(\omega_e,\omega_f) \,\partial_iJ_e(x)\,\partial_jJ_f(x)\geq0. \end{align*} Indeed, all first derivatives of the couplings are nonnegative, and their mixed derivatives at distinct coordinates vanish except for \(e=\{i,j\}\), where \[ \partial_{ij}J_e(x)=\frac{J_e(x)}{4x_ix_j}\geq0. \] Lemma~\ref{lem:angular} supplies the remaining nonnegative factors. \end{proof} \subsection{Step 4, squarefree moments} The MTP\(_2\) property proves the Koteljanskii inequality~\eqref{eq:lsm} after a change of measure, as shown below. % \begin{lemma}\label{lem:moments} Suppose that a random vector \(X\in(0,\infty)^n\) has an MTP\(_2\) density and finite squarefree moments. Then, the set function \(f(S)=\E\prod_{i\in S}X_i\) is log-supermodular. \end{lemma} \begin{proof} Fix \(S,T\) and put \(R=S\cap T\). The density $p_R(x)=f(R)^{-1}\prod_{i\in R}x_ip(x)$ is MTP\(_2\), since the multiplying factor is a product of functions of individual coordinates. By the FKG inequality for MTP\(_2\) densities \citep{karlin1980classes}, increasing functions are positively correlated under this density. Apply this rule to the increasing functions \(u(x)=\prod_{i\in S\setminus R}x_i\) and \(v(x)=\prod_{i\in T\setminus R}x_i\), first truncated at a level \(N\) and then with \(N\to\infty\) by monotone convergence; the assumed moments make all terms finite. It yields \[ \frac{f(S\cup T)}{f(R)} \geq \frac{f(S)}{f(R)}\frac{f(T)}{f(R)}, \] which proves the assertion. \end{proof} \noindent Now we have all the steps, and can now state the final proof. \begin{proof}[\textbf{Proof of Theorem~\ref{thm:main}}] By Step~1 we may assume that \(A\) is symmetric. By Step~3's Proposition~\ref{prop:radial}, the vector \(X\) has an MTP\(_2\) density, while by Step~2 its squarefree moments are \(\per(A_S)\). Thus, using Step~4's Lemma~\ref{lem:moments} we may immediately conclude~\eqref{eq:lsm}. \end{proof} \begin{rmk}[Repeated indices]\label{rmk:mixed} The same proof gives log-supermodularity of all mixed moments of \(X\) on \(\mathbb N_0^n\): tilt by \(x^{a\wedge b}\) and apply the FKG inequality to \(x^{a-a\wedge b}\) and \(x^{b-a\wedge b}\). By the same moment theorem \citep{reed1962moment}, these moments are the permanents of the matrices obtained from \(A\) by repeating row and column \(i\) the same number of times. \end{rmk} \begin{rmk}[Positive association is not enough]\label{rmk:assoc} The tilt in the proof of Lemma~\ref{lem:moments} cannot be skipped: positive association of a random vector does not make its squarefree moments log-supermodular. For a random vector \(W\in\reals_{\ge0}^n\) write \(\Phi_W(S)=\E\prod_{i\in S}W_i\). For independent \(N\sim\operatorname{Poisson}(\lambda)\) and \(N'\sim\operatorname{Poisson}(\mu)\), a constant \(a\ge0\), and \(W=(a+N,a+N+N',a+N')\), a direct moment calculation gives \[ \Phi_W(\{1,2,3\})\Phi_W(\{2\})-\Phi_W(\{1,2\})\Phi_W(\{2,3\})=-\lambda\mu. \] The vector \(W\) is positively associated, since its coordinates are nondecreasing functions of the independent variables \(N\) and \(N'\) \citep{esary1967association}. The same example, with \(a=0\) and \(\lambda=\mu=1\), shows that disjoint convolution does not preserve log-supermodularity. Here \(\Phi_W(\{1,2,3\})\Phi_W(\{2\})=4\cdot2<3\cdot3=\Phi_W(\{1,2\})\Phi_W(\{2,3\})\). Since \(W\) is the sum of the independent vectors \((N,N,0)\) and \((0,N',N')\), expanding \(\prod_{i\in S}W_i\) over the two summands shows that \(\Phi_W\) is the disjoint convolution \[ (f\dotast g)(S):=\sum_{R\subseteq S}f(R)g(S\setminus R) \] of their moment functions \(f\) and \(g\). Both are log-supermodular: \(f\) vanishes outside \(2^{\{1,2\}}\), where the only nontrivial instance is \(f(\{1,2\})f(\varnothing)=2\ge1=f(\{1\})f(\{2\})\), and similarly for \(g\) on \(2^{\{2,3\}}\). The example therefore contradicts Corollary~11 of the technical report of \citet{lovasz2003}, which asserts that the disjoint convolution of two log-supermodular functions is log-supermodular. \end{rmk} \section{Counterexamples}\label{sec:counter} This section proves Theorem~\ref{thm:counterexample}; it also briefly considers the more general setup of \(\alpha\)-permanents, where a similar negative result holds (Theorem~\ref{thm:allalpha}). Both results come from inverse \(M\)-matrices whose inverses have the directed graph of two cycles that share a path; the matrix of Theorem~\ref{thm:counterexample} belongs, after a column scaling, to the family used for Theorem~\ref{thm:allalpha} (Remark~\ref{rmk:family}). \subsection{Proof of Theorem~\ref{thm:counterexample}} \begin{proof}[\textbf{Proof of Theorem~\ref{thm:counterexample}}] Direct multiplication gives \[ A^{-1}=\frac1{84}\begin{pmatrix} 7&0&-7&0&0\\ 0&7&-7&0&0\\ 0&0&14&-14&0\\ 0&0&0&14&-14\\ -1&-1&0&0&14 \end{pmatrix}. \] This matrix has nonpositive off-diagonal entries and inverse \(A>0\). The matrix \(A\) is not symmetrizable: if \(\Delta A\Delta^{-1}\) were symmetric, so would be its inverse \(\Delta A^{-1}\Delta^{-1}\), but a diagonal similarity preserves the zero pattern of \(A^{-1}\), and \((A^{-1})_{13}\ne0=(A^{-1})_{31}\). Take \(S=\{1,3,4,5\}\), \(T=\{2,3,4,5\}\), and \(R=\{3,4,5\}\). The four permanents are \[ \per(A)=182784,\qquad \per(A_R)=546,\qquad \per(A_S)=\per(A_T)=9996. \] Consequently, \[ \per(A)\per(A_R)=99800064 <99920016=\per(A_S)\per(A_T), \] and their ratio is \(832/833\). \end{proof} \subsection{\texorpdfstring{\(\alpha\)}{alpha}-permanents}\label{sec:alpha} While developing Theorems~\ref{thm:main} and~\ref{thm:counterexample} we wondered whether similar results might hold for the \(\alpha\)-permanents of \citet{verejones1997alpha}. The situation here turns out to be more nuanced, as noted in Remark~\ref{rmk:real} below. For \(\alpha>0\), the \(\alpha\)-permanent of a matrix \(H\) indexed by \(S\) is \begin{equation}\label{eq:alphaper} \per_\alpha(H)=\sum_{\pi\in\mathfrak S(S)}\alpha^{c(\pi)}\prod_{i\in S}H_{i,\pi(i)}, \end{equation} where \(\mathfrak S(S)\) is the group of permutations of \(S\) and \(c(\pi)\) is the number of cycles of \(\pi\); the ordinary permanent is \(\alpha=1\). A random vector \(Y\in[0,\infty)^n\) with Laplace transform \(\det(I+A\Lambda)^{-\alpha}\), where \(\Lambda=\diag(s_1,\dots,s_n)\), is an \(\alpha\)-permanental vector with kernel \(A\) in the sense of \citet{verejones1997alpha}, and its squarefree moments are the principal \(\alpha\)-permanents, \begin{equation}\label{eq:vj} \E\prod_{i\in S}Y_i=\per_\alpha(A_S),\qquad S\subseteq[n], \end{equation} by \citet[Proposition~4.2]{verejones1997alpha}. \citet{eisenbaum2009permanental} proved that an infinitely divisible permanental vector with kernel \(A\) exists if and only if \(A\) is, up to diagonal similarity, the Green function of a transient Markov chain, that is, an inverse \(M\)-matrix. \begin{rmk}[Positive cases]\label{rmk:real} For symmetric positive definite \(A\) there is a real Gaussian analogue. If \(\eta\sim N(0,A)\), then \((\eta_i^2)_{i}\) is MTP\(_2\) exactly when \(A^{-1}\) is, up to a signature, an \(M\)-matrix \citep{karlin1981total}. The vector \((\eta_i^2/2)_i\) has Laplace transform \(\det(I+A\Lambda)^{-1/2}\), so by~\eqref{eq:vj} its squarefree moments are \(\per_{1/2}(A_S)\), and \(\E\prod_{i\in S}\eta_i^2=2^{|S|}\per_{1/2}(A_S)\). Lemma~\ref{lem:moments} and the diagonal similarity argument above therefore give Theorem~\ref{thm:main} for \(\per_{1/2}\), but not for the permanent itself. If \(A\) is a symmetric inverse \(M\)-matrix and the graph of \(A^{-1}\) is a tree, \citet[Section~4]{royen2025improved} proves that the \(\alpha\)-permanental vector with kernel \(A\) has an MTP\(_2\) density for every \(\alpha>0\); formula~\eqref{eq:vj} and Lemma~\ref{lem:moments} then give log-supermodularity of \(S\mapsto\per_\alpha(A_S)\) for every \(\alpha>0\). \end{rmk} The Gaussian argument uses \(\alpha=1\) (complex) and \(\alpha=\tfrac12\) (real) in an essential way, and for other \(\alpha\) the symmetrizable case is open when the graph of \(A^{-1}\) has cycles (Section~\ref{sec:open}). For general inverse \(M\)-matrices the answer is negative. Every inverse \(M\)-matrix is the kernel of an \(\alpha\)-permanental vector for every \(\alpha>0\) \citep{eisenbaum2009permanental}, so one may ask whether some other order \(\alpha\) gives the analogue of~\eqref{eq:lsm} for all inverse \(M\)-matrices. None does---see Appendix~\ref{app:alpha} for details. \section{Open problems}\label{sec:open} \begin{enumerate}[label=(\alph*),itemsep=1pt] \item Characterize the inverse \(M\)-matrices whose principal permanents are log-supermodular. By Theorem~\ref{thm:main} this class contains the symmetrizable ones, and by Theorem~\ref{thm:counterexample} it is smaller than the class of all inverse \(M\)-matrices. \item For symmetrizable inverse \(M\)-matrices and \(\alpha\notin\{\tfrac12,1\}\), decide whether \(S\mapsto\per_\alpha(A_S)\) is log-supermodular. Theorem~\ref{thm:main} and Remark~\ref{rmk:real} settle \(\alpha=1\), \(\alpha=\tfrac12\), and every \(\alpha>0\) when the graph of \(A^{-1}\) is a tree. \item The proof of Theorem~\ref{thm:allalpha} needs the dimension to grow with \(\alpha\). For fixed \(n\), is there an \(\alpha_n\) such that \(S\mapsto\per_\alpha(A_S)\) is log-supermodular for every \(n\times n\) inverse \(M\)-matrix and every \(\alpha\ge\alpha_n\)? \item For an inverse \(M\)-matrix \(A\), consider \[ f_A(x_0,x)=\nlsum_{S\subseteq[n]}\per(A_S)^{-1}x_0^{\,n-|S|}\prod\nolimits_{i\in S}x_i. \] For positive semidefinite \(A\), the polynomial with coefficients \(\det(A_S)\) in place of \(1/\per(A_S)\) is \(\det(x_0I+\diag(x)A)\), which is real stable \citep{branden08} and hence Lorentzian \citep{branden2020lorentzian}. Is \(f_A\) Lorentzian for every inverse \(M\)-matrix \(A\)? Numerical tests with \(n\le5\) found no counterexample, including the matrix of Theorem~\ref{thm:counterexample}; for random nonnegative \(A\), by contrast, \(f_A\) is usually not Lorentzian, and for symmetric inverse \(M\)-matrices the polynomial with coefficients \(\per(A_S)\) was never Lorentzian. \end{enumerate} \section*{AI Usage} The conjecture was formulated by the author in April 2015, with partial proofs. After a sequence of back and forth discussions with GPT-5.5 (Pro) the first proof was attained. Subsequently, an audit with GPT-6 (Astra) revealed a few small errors, that were fixed. AI assisted with the drafting of the paper (GPT-6, and Claude Opus 5.5 and Fable 5.1), while the author strove to rewrite the entire paper by hand to simplify, polish, and compactify. \section*{Acknowledgements} The author (SS) thanks the Alexander von Humboldt (AvH) Foundation for their generous support via a Humboldt Professorship in AI. \begin{thebibliography}{23} \providecommand{\natexlab}[1]{#1} \providecommand{\url}[1]{\texttt{#1}} \expandafter\ifx\csname urlstyle\endcsname\relax \providecommand{\doi}[1]{doi: #1}\else \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi \bibitem[Berman and Plemmons(1994)]{berman1994} Abraham Berman and Robert~J. Plemmons. \newblock \emph{Nonnegative Matrices in the Mathematical Sciences}. \newblock Classics in Applied Mathematics. SIAM, Philadelphia, PA, 1994. \newblock \doi{10.1137/1.9781611971262}. \bibitem[Borcea and Br{\"a}nden(2008)]{branden08} Julius Borcea and Petter Br{\"a}nden. \newblock Applications of stable polynomials to mixed determinants: {J}ohnson's conjectures, unimodality, and symmetrized {F}ischer products. \newblock \emph{Duke Mathematical Journal}, 143\penalty0 (2):\penalty0 205--223, 2008. \bibitem[Br{\"a}nd{\'e}n and Huh(2020)]{branden2020lorentzian} Petter Br{\"a}nd{\'e}n and June Huh. \newblock Lorentzian polynomials. \newblock \emph{Annals of Mathematics}, 192\penalty0 (3):\penalty0 821--891, 2020. \newblock \doi{10.4007/annals.2020.192.3.4}. \bibitem[Eisenbaum(2014)]{eisenbaum2014characterization} Nathalie Eisenbaum. \newblock Characterization of positively correlated squared {G}aussian processes. \newblock \emph{The Annals of Probability}, 42\penalty0 (2):\penalty0 559--575, 2014. \newblock \doi{10.1214/12-AOP807}. \bibitem[Eisenbaum(2022)]{eisenbaum2022erratum} Nathalie Eisenbaum. \newblock Erratum to ``{C}haracterization of positively correlated squared {G}aussian processes''. \newblock \emph{The Annals of Probability}, 50\penalty0 (4), 2022. \newblock \doi{10.1214/21-AOP1561}. \bibitem[Eisenbaum and Kaspi(2009)]{eisenbaum2009permanental} Nathalie Eisenbaum and Haya Kaspi. \newblock On permanental processes. \newblock \emph{Stochastic Processes and their Applications}, 119\penalty0 (5):\penalty0 1401--1415, 2009. \newblock \doi{10.1016/j.spa.2008.07.003}. \bibitem[Esary et~al.(1967)Esary, Proschan, and Walkup]{esary1967association} James~D. Esary, Frank Proschan, and David~W. Walkup. \newblock Association of random variables, with applications. \newblock \emph{The Annals of Mathematical Statistics}, 38\penalty0 (5):\penalty0 1466--1474, 1967. \newblock \doi{10.1214/aoms/1177698701}. \bibitem[Fallat et~al.(1998)Fallat, Hall, and Johnson]{fallat1998characterization} Shaun~M. Fallat, H.~Tracy Hall, and Charles~R. Johnson. \newblock Characterization of product inequalities for principal minors of {$M$}-matrices and inverse {$M$}-matrices. \newblock \emph{The Quarterly Journal of Mathematics}, 49\penalty0 (4):\penalty0 451--458, 1998. \newblock \doi{10.1093/qmathj/49.4.451}. \bibitem[Finner and Roters(2024)]{finner2024positive} Helmut Finner and Markus Roters. \newblock On positive association of absolute-valued and squared multivariate {G}aussians beyond {MTP}$_2$. \newblock \emph{Journal of Multivariate Analysis}, 202:\penalty0 Art.~105295, 2024. \newblock \doi{10.1016/j.jmva.2024.105295}. \bibitem[Ginibre(1970)]{ginibre1970general} Jean Ginibre. \newblock General formulation of {G}riffiths' inequalities. \newblock \emph{Communications in Mathematical Physics}, 16\penalty0 (4):\penalty0 310--328, 1970. \newblock \doi{10.1007/BF01646537}. \bibitem[Johnson(1982)]{johnson1982inverse} Charles~R. Johnson. \newblock Inverse {$M$}-matrices. \newblock \emph{Linear Algebra and its Applications}, 47:\penalty0 195--216, 1982. \newblock \doi{10.1016/0024-3795(82)90238-5}. \bibitem[Johnson and Smith(2011)]{johnson2011inverse} Charles~R. Johnson and Ronald~L. Smith. \newblock Inverse {$M$}-matrices, {II}. \newblock \emph{Linear Algebra and its Applications}, 435\penalty0 (5):\penalty0 953--983, 2011. \newblock \doi{10.1016/j.laa.2011.02.016}. \bibitem[Karlin and Rinott(1980)]{karlin1980classes} Samuel Karlin and Yosef Rinott. \newblock Classes of orderings of measures and related correlation inequalities. {I}. {M}ultivariate totally positive distributions. \newblock \emph{Journal of Multivariate Analysis}, 10\penalty0 (4):\penalty0 467--498, 1980. \newblock \doi{10.1016/0047-259X(80)90065-2}. \bibitem[Karlin and Rinott(1981)]{karlin1981total} Samuel Karlin and Yosef Rinott. \newblock Total positivity properties of absolute value multinormal variables with applications to confidence interval estimates and related probabilistic inequalities. \newblock \emph{The Annals of Statistics}, 9\penalty0 (5):\penalty0 1035--1049, 1981. \newblock \doi{10.1214/aos/1176345583}. \bibitem[Koteljanskii(1950)]{koteljanskii1950theory} D.~M. Koteljanskii. \newblock The theory of nonnegative and oscillating matrices. \newblock \emph{Ukrainski\u{\i} Matematicheski\u{\i} Zhurnal}, 2:\penalty0 94--101, 1950. \newblock In Russian. English translation: Amer.\ Math.\ Soc.\ Transl.\ (2) \textbf{27} (1963), 1--8. \bibitem[Koteljanskii(1953)]{koteljanskii1953property} D.~M. Koteljanskii. \newblock A property of sign-symmetric matrices. \newblock \emph{Uspekhi Matematicheskikh Nauk (N.S.)}, 8:\penalty0 163--167, 1953. \newblock In Russian. English translation: Amer.\ Math.\ Soc.\ Transl.\ (2) \textbf{27} (1963), 19--24. \bibitem[Lieb(1966)]{lieb1966} Elliott~H. Lieb. \newblock Proofs of some conjectures on permanents. \newblock \emph{Journal of Mathematics and Mechanics}, 16\penalty0 (2):\penalty0 127--134, 1966. \newblock \doi{10.1512/iumj.1967.16.16008}. \bibitem[Lov{\'a}sz and Saks(2003)]{lovasz2003} L{\'a}szl{\'o} Lov{\'a}sz and Michael Saks. \newblock Discrete localization and correlation inequalities for set functions. \newblock Technical Report MSR-TR-2003-07, Microsoft Research, 2003. \newblock \url{https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/tr-2003-07.pdf}. \bibitem[Marcus(1963)]{marcus1963permanent} Marvin Marcus. \newblock The permanent analogue of the {H}adamard determinant theorem. \newblock \emph{Bulletin of the American Mathematical Society}, 69\penalty0 (4):\penalty0 494--497, 1963. \newblock \doi{10.1090/S0002-9904-1963-10975-1}. \bibitem[Reed(1962)]{reed1962moment} Irving~S. Reed. \newblock On a moment theorem for complex {G}aussian processes. \newblock \emph{IRE Transactions on Information Theory}, 8\penalty0 (3):\penalty0 194--195, 1962. \newblock \doi{10.1109/TIT.1962.1057719}. \bibitem[Royen(2025)]{royen2025improved} Thomas Royen. \newblock On improved {G}aussian correlation inequalities for symmetrical $n$-rectangles extended to certain multivariate gamma distributions and some further probability inequalities. \newblock \emph{Far East Journal of Theoretical Statistics}, 69\penalty0 (1):\penalty0 1--38, 2025. \newblock \doi{10.17654/0972086325001}. \bibitem[Skandera and Soskin(2025)]{skandera2025permanental} Mark Skandera and Daniel Soskin. \newblock Permanental inequalities for totally positive matrices. \newblock \emph{The Electronic Journal of Combinatorics}, 32\penalty0 (1):\penalty0 P1.14, 2025. \newblock \doi{10.37236/13098}. \bibitem[Vere-Jones(1997)]{verejones1997alpha} David Vere-Jones. \newblock Alpha-permanents and their applications to multivariate gamma, negative binomial and ordinary binomial distributions. \newblock \emph{New Zealand Journal of Mathematics}, 26:\penalty0 125--149, 1997. \end{thebibliography} \appendix \section{More on $\alpha$-permanents} \label{app:alpha} \begin{theorem}\label{thm:allalpha} For every \(\alpha>0\) there are an integer \(n\), an entrywise positive \(n\times n\) inverse \(M\)-matrix \(A\), and sets \(S,T\subseteq[n]\) such that \[ \per_\alpha(A_{S\cup T})\per_\alpha(A_{S\cap T}) <\per_\alpha(A_S)\per_\alpha(A_T). \] \end{theorem} \begin{proof}[\textbf{Proof of Theorem~\ref{thm:allalpha}}] Fix \(k\geq1\) and take the \(n=k+2\) vertices \(1,2,r_1,\ldots,r_k\). Let \(B\) have weight one on the edges \(1\to r_1\), \(2\to r_1\), and \(r_j\to r_{j+1}\), and weight \(t\) on \(r_k\to1\) and \(r_k\to2\), with \(0