Digital Commons Network^™

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's exposition of Egorychev's proof for the van der Waerden's conjecture on the permanents of doubly stochastic matrices. The purpose of this component is to provide elementary proofs of several interesting known …