Van der Waerden’s permanent conjecture

Let $A$ be any doubly stochastic $n\times n$ matrix (i.e. nonnegative real entries, each row sums to 1, each column too, hence square).

Let $A^{\circ }$ be the one where all entries are equal (i.e. they are $\frac{1}{n}$). Its permanent works out to

$$per{A}^{\circ }=n!{(\frac{1}{n})}^n$$

and Van der Waerden conjectured in 1926 that this is the smallest value for the permanent of any doubly stochastic $A$, and is attained only for $A=A^{\circ }$:

$$perA>n!{(\frac{1}{n})}^n\mathit{\hspace{1em}}\text{(for }A\ne A^{\circ }\text{).}$$

It was finally proven independently by Egorychev and by Falikman, in 1979/80.