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.
References
 1

Hal86
Marshall J. Hall, Jr.,
Combinatorial Theory (2nd ed.),
Wiley 1986, repr. 1998, ISBN 0 471 09138 3 and 0 471 31518 4
has a proof of the permanent conjecture.
Title  Van der Waerden’s permanent conjecture 

Canonical name  VanDerWaerdensPermanentConjecture 
Date of creation  20130322 15:10:51 
Last modified on  20130322 15:10:51 
Owner  marijke (8873) 
Last modified by  marijke (8873) 
Numerical id  5 
Author  marijke (8873) 
Entry type  Theorem 
Classification  msc 15A15 
Classification  msc 15A51 
Synonym  permanent conjecture 