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:
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:
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: