Notes on the proof of the van der Waerden permanent conjecture

The permanent of an $n\times n$ matrix $A=(a_{i j})$ with real entries is defined by the sum $$\sum_{\sigma \in S_n} \prod_{i=1}^{n} a_{i \sigma(i)}$$ where $S_n$ denotes the symmetric group on the $n$-element set $\{1,2,\dots,n\}$. In this creative component we survey some known properties of permanents, calculation of permanents for particular types of matrices and their applications in combinatorics and linear algebra. Then we follow the lines of van Lint\u27s exposition of Egorychev\u27s proof for the van der Waerden\u27s conjecture on the permanents of doubly stochastic matrices. The purpose of this component is to provide elementary proofs of several interesting known facts related to permanents of some special matrices. It is an expository survey paper in nature and reports no new findings

Matrix permanent and quantum entanglement of permutation invariant states

We point out that a geometric measure of quantum entanglement is related to the matrix permanent when restricted to permutation invariant states. This connection allows us to interpret the permanent as an angle between vectors. By employing a recently introduced permanent inequality by Carlen, Loss and Lieb, we can prove explicit formulas of the geometric measure for permutation invariant basis states in a simple way.Comment: 10 page

A simple polynomial time algorithm to approximate the permanent within a simply exponential factor

We present a simple randomized polynomial time algorithm to approximate the mixed discriminant of $n$ positive semidefinite $n \times n$ matrices within a factor $2^{O(n)}$. Consequently, the algorithm allows us to approximate in randomized polynomial time the permanent of a given $n \times n$ non-negative matrix within a factor $2^{O(n)}$. When applied to approximating the permanent, the algorithm turns out to be a simple modification of the well-known Godsil-Gutman estimator