Decomposability of Weak Majorization

Let x, y\in \mathbb{R}^n. We use the notation x\prec_w y when x is weakly majorized by y. We say that x\prec_w y is decomposable at k (۱\leq k < n) if x\prec_w y has a coincidence at k and y_{k}\neq y_{k+۱}. Corresponding to this majorization we have a doubly substochastic matrix P. The paper presents x\prec_w y is decomposable at some k (۱\leq k