Let $A \in \Omega_n$ be doubly-stochastic $n \times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality

\begin{equation} \label{le}

per(\widetilde{A}) \geq \prod_{1 \leq i,j \leq n} (1- A(i,j)); \widetilde{A}(i,j) =: A(i,j)(1-A(i,j)), 1 \leq i,j \leq n

\end{equation}

We prove in this paper the following generalization (or just clever reformulation) of (\ref{le}):\\

For all pairs of $n \times n$ matrices $(P,Q)$, where $P$ is nonnegative and $Q$ is doubly-stochastic

\begin{equation} \label{st}

\log(per(P)) \geq \sum_{1 \leq i,j \leq n} \log(1- Q(i,j)) (1- Q(i,j)) - \sum_{1 \leq i,j \leq n} Q(i,j) \log \left(\frac{Q(i,j)}{P(i,j)} \right

)

\end{equation}

The main co

rollary of (\ref{st}) is the following inequality for doubly-stochastic matrices:

$$

\frac{per(A)}{F(A)} \geq 1; F(A) =: \prod_{1 \leq i,j \leq n} \left(1- A(i,j)\right)^{1- A(i,j)}.

$$

{\bf We use this inequality to prove Friedland's conjecture on monomer-dimer entropy, so called {\it Asymptotic Lower Matching Conjecture}}\\

We present explicit doubly-stochastic $n \times n$ matrices $A$ with

the ratio $\frac{per(A)}{F(A)} = \sqrt{2}^{n}$ and conjecture that

$$

\max_{A \in \Omega_n}\frac{per(A)}{F(A)} \approx \left(\sqrt{2} \right)^{n}.

$$

If true, it would imply a deterministic poly-time algorithm to approximate the permanent of $n \times n$ nonnegative

matrices within the relative factor $\left(\sqrt{2} \right)^{n}$.\\

A lot of editing, a new section on correlational inequalities

is added.

Let $A \in \Omega_n$ be doubly-stochastic $n \times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality

\begin{equation} \label{le}

per(\widetilde{A}) \geq \prod_{1 \leq i,j \leq n} (1- A(i,j)); \widetilde{A}(i,j) =: A(i,j)(1-A(i,j)), 1 \leq i,j \leq n

\end{equation}

We prove in this paper the following generalization (or just clever reformulation) of (\ref{le}):\\

For all pairs of $n \times n$ matrices $(P,Q)$, where $P$ is nonnegative and $Q$ is doubly-stochastic

\begin{equation} \label{st}

\log(per(P)) \geq \sum_{1 \leq i,j \leq n} \log(1- Q(i,j)) (1- Q(i,j)) - \sum_{1 \leq i,j \leq n} Q(i,j) \log \left(\frac{Q(i,j)}{P(i,j)} \right

)

\end{equation}

The main co

rollary of (\ref{st}) is the following inequality for doubly-stochastic matrices:

$$

\frac{per(A)}{F(A)} \geq 1; F(A) =: \prod_{1 \leq i,j \leq n} \left(1- A(i,j)\right)^{1- A(i,j)}.

$$

{\bf We use this inequality to prove Friedland's conjecture on monomer-dimer entropy, so called {\it Asymptotic Lower Matching Conjecture}(LAMC)\\

We use some ideas of our proof of (LAMC) to disprove [Lu,Mohr,Szekely]

positive correlation conjecture.

We present explicit doubly-stochastic $n \times n$ matrices $A$ with

the ratio $\frac{per(A)}{F(A)} = \sqrt{2}^{n}$ and conjecture that

$$

\max_{A \in \Omega_n}\frac{per(A)}{F(A)} \approx \left(\sqrt{2} \right)^{n}.

$$

If true, it would imply a deterministic poly-time algorithm to approximate the permanent of $n \times n$ nonnegative

matrices within the relative factor $\left(\sqrt{2} \right)^{n}$.\\

A section on monomer-dimer problem is seriously revised;

a new section with a disproof of

[Lu,Mohr,Szekely] is added. The current version is

longer and cleaner.

Let $A \in \Omega_n$ be doubly-stochastic $n \times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality

\begin{equation} \label{le}

per(\widetilde{A}) \geq \prod_{1 \leq i,j \leq n} (1- A(i,j)); \widetilde{A}(i,j) =: A(i,j)(1-A(i,j)), 1 \leq i,j \leq n

\end{equation}

We prove in this paper the following generalization (or just clever reformulation) of (\ref{le}):\\

For all pairs of $n \times n$ matrices $(P,Q)$, where $P$ is nonnegative and $Q$ is doubly-stochastic

\begin{equation} \label{st}

\log(per(P)) \geq \sum_{1 \leq i,j \leq n} \log(1- Q(i,j)) (1- Q(i,j)) - \sum_{1 \leq i,j \leq n} Q(i,j) \log \left(\frac{Q(i,j)}{P(i,j)} \right

)

\end{equation}

The main co

rollary of (\ref{st}) is the following inequality for doubly-stochastic matrices:

$$

\frac{per(A)}{F(A)} \geq 1; F(A) =: \prod_{1 \leq i,j \leq n} \left(1- A(i,j)\right)^{1- A(i,j)}.

$$

{\bf We use this inequality to prove Friedland's conjecture on monomer-dimer entropy, so called {\it Asymptotic Lower Matching Conjecture}}\\

We present explicit doubly-stochastic $n \times n$ matrices $A$ with

the ratio $\frac{per(A)}{F(A)} = \sqrt{2}^{n}$ and conjecture that

$$

\max_{A \in \Omega_n}\frac{per(A)}{F(A)} \approx \left(\sqrt{2} \right)^{n}.

$$

If true, it would imply a deterministic poly-time algorithm to approximate the permanent of $n \times n$ nonnegative

matrices within the relative factor $\left(\sqrt{2} \right)^{n}$.\\