filling up numbers in a matrix Suppose you have a $k.n \times 2$ matrix. You have to fill up the numbers $1,2,3, \cdots, n$ as entries in such a way that in each column it is non-decreasing, in each row it is strictly increasing and each number should appear exactly $2k$ times. How many ways one can fill up the matrix ?  

There are no ways to fill out a matrix with these constraints.  Think about where the 1's in the first column must go.  They must be first, because they cannot come after any other number.  So the first k entries in the first column are 1's.  What can go to the right of those?  It must be a 2 or larger.  But, where will the one's go in the second column?  Anywhere you put them will remove the non-decreasing requirement on the second column.  So it cannot be done.
