On a conjecture concerning the Bruhat order

Let R and S be two sequences of positive integers in nonincreasing order having the same sum. Let A(R,S) be the class of all (0,1)-matrices with row sum vector R and column sum vector S. If A(R,S) is nonempty, an inversion in A∈A(R,S) consists of two entries of A equal to 1, one of them is located to the top-right of the other. Let ν(A) be the total number of inversions in A. The Bruhat order is a partial order defined on A(R,S) and denoted by ⪯B. In this paper, we prove the conjecture:•“If A,C∈A(R,S), A≠C and A⪯BC then ν(A)