building a matrix with conditions for rows and columns

I want to build a matrix in the following manner

Enter number of switching elements – “k”
Enter number of coupling edges- “n”
Total structures possible= fact (n+k-1)/ (fact (k)*fact (n-1))

Total number of columns=n
Total number of rows= All possible combinations with sum=k

For example

n=10
k=9
Total Structures=48620

e12  e13  e14  e15  e23  e24  e25  e34  e35  e45
0  0  0  0  0  0  0  0  0  9
0  0  0  0  0  0  0  0  1  8
0  0  0  0  0  0  0  0  2  7
0  0  0  0  0  0  0  0  3  6
0  0  0  0  0  0  0  0  4  5
0  0  0  0  0  0  0  0  5  4
0  0  0  0  0  0  0  0  6  3
0  0  0  0  0  0  0  0  7  2
0  0  0  0  0  0  0  0  8  1
0  0  0  0  0  0  0  0  9  0
0  0  0  0  0  0  0  1  0  8
0  0  0  0  0  0  0  1  1  7
.
.
.

So on till I have this 48620*10 matrix in the same manner as shown above. I then have to perform calculations with this matrix as well

BR Manoj