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On the Van Der Waerden conjecture for matrices of rank two

The generalized van der Waerden conjecture asserts that Here Ais any doubly stochastic n×n matrixJn is the n×n matrix all whose entries are equal to 1/n and pk (A) is the sum of the permanents of all k×k submatrices of A. The casek=n of (*) is the (ordinary) van der Waerden conjecture.It is proved that if the van der Waerden conjecture holds for all k×k k⩽n doubly stochastic matrices of rank two, then the generalized conjecture also holds for the same matrices. The proof is based on a representation of pk (A) by means of an associated pair of polynomials.