The monotonicity of and the Đoković conjectures on permanents of doubly stochastic matrices (Q1077492)

From author's summary: Let \(\Omega_ n\) denote the set of all \(n\times n\) doubly stochastic matrices, and let \(J_ n=[1/n]_{n\times n}.\) For \(A\in \Omega_ n\) and any integer \(k\), \(1\leq k\leq n\), let \(p_ k(A)\) denote the sum of all subpermanents of order \(k\) of \(A\), and let \(D_ k(A)=p_ k(A)-[(n-k+1)^ 2/nk]p_{k-1}(A)\) for \(k=2,...,n\). \(A\in \Omega_ n\) is called a \(D_ k\)-minimizing matrix on \(\Omega_ n\) if \(D_ k(A)\leq D_ k(X)\) for every \(X\in \Omega_ n\). The Đoković conjecture asserts that if \(A\) is a \(D_ k\)-minimizing matrix on \(\Omega_ n\), then \(D_ k(A)=0\) \((k=2,...,n)\). In this paper, we prove that if \(A\) is a positive \(D_ k\)-minimizing matrix on \(\Omega_ n\), then \(D_ k(A)=0\) and \(A=J_ n\) \((k=2,...,n)\), and we settle the conjecture for an \(n-2\) dimensional face of \(\Omega_ n\). We also prove the monotonicity of the permanent for \(A=\left[ \begin{matrix} X\\ U\end{matrix} \begin{matrix} Y\\ V\end{matrix} \right]\in \Omega_ n\) where each of the blocks of \(A\) is a matrix of equal entries, and for any \(A\in \Omega_ n\) with \(n-1\) identical rows.