Let $0\le x_1\le x_2\le \cdots\le x_n\le n-1$ be given. My question is as follows : Under which condition there exists a doubly stochastic matrix $M=(m_{i,j})_{1\le i,j\le n}$ s.t.

$$\sum_{j=1}^n (j-1)m_{i,j}=x_i \quad \mbox{for all } 1\le i\le n,$$

$$\sum_{j=k}^nm_{i,j}\le \sum_{j=k}^nm_{i+1,j},\quad \mbox{for all } 1\le i\le n-1 \mbox{ and } 1\le k\le n.$$

Here a double stochastic matrix refers to $\sum_{j=1}^n m_{i,j}=1$ for all $1\le i\le n$ and $\sum_{i=1}^n m_{i,j}=1$ for all $1\le j\le n$. Any answers or comments are highly appreciated!

Let $0\le x_1\le x_2\le \cdots\le x_n\le n-1$ be given. My question is as follows : Under which condition there exists a doubly stochastic matrix $M=(m_{i,j})_{1\le i,j\le n}$ s.t.

$$\sum_{j=1}^n (j-1)m_{i,j}=x_i \quad \mbox{for all } 1\le i\le n,$$

$$\sum_{j=k}^nm_{i,j}\le \sum_{j=k}^nm_{i+1,j},\quad \mbox{for all } 1\le i\le n-1 \mbox{ and } 1\le k\le n.$$

Here a double stochastic matrix refers to $\min_{1\le i,j\le n}m_{i,j}\ge 0$, $\sum_{j=1}^n m_{i,j}=1$ for all $1\le i\le n$ and $\sum_{i=1}^n m_{i,j}=1$ for all $1\le j\le n$. Any answers or comments are highly appreciated!