Existence of a specific stochastic matrix 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!
 A: Let $$x = \left[\begin{matrix} x_1\\ x_2 \\ \vdots \\ x_n\end{matrix}\right] \quad \textrm{and} \quad y = \left[\begin{matrix}0 \\ 1 \\ \vdots \\ n-1\end{matrix}\right] $$
Your first condition, that there exists a doubly stochastic matrix $M=[m_{i,j}]$
$$
\sum_{j=1}^n (j-1)m_{i,j} = x_i \quad \textrm{for all}\ 1\leq i\leq n
$$
becomes $My = x$ which by the Hardly-Littlewood-Polya Theorem is equivalent to $y \prec x$, or $y$ is majorized by $x$. In other words,
$$
\sum_{j=k}^n (j-1) \leq \sum_{j=k}^n x_j \quad \textrm{for all} \ 1\leq k\leq n \quad \textrm{and} \quad \sum_{j=1}^n (j-1) = \sum_{j=1}^n x_j
$$
In particular, one can choose $M$ such that $JMJ$, where
$$
J = \left[\begin{matrix} &&1 \\ & \unicode{x22f0} & \\ 1&&\end{matrix}\right],
$$
is a uniformly tapered doubly stochastic matrix:
$$
m_{1,1} \geq m_{2,1} \geq \cdots \geq m_{1,n}, \ m_{1,n} \leq m_{2,n} \leq \cdots \leq m_{n,n}, \quad \textrm{and} \quad m_{i,j} + m_{i+1,j-1} \geq m_{i,j-1} + m_{i+1,j} 
$$
for $2\leq i, j \leq n-1$. See Defintion 2.B.5 and Theorem 2.B.6 in Marshall and Olkin's book.
This gets you some way towards your second set of inequalities on the columns of $M$, for instance, the $2\times 2$ case is satisfied. However, I am not sure if you can prove your inequalities from this in general.
