Multinomial matrix

I want to generate a multinomial matrix such that with given K(interger)>0 I can get a matrix with all the possible combination of K+1 variables whose sum is from 1 to K eg for K=3, we have x1, x2, x3, x4 variables

we should have

1 0 0 0

...

0 0 0 1

2 0 0 0

...

0 0 0 2

1 1 0 0

...

3 0 0 0

...

1 1 0 1

There should be total 34 rows/combination it is impossible to manually input if k increases I wonder if anyone can suggest a proper way/function to generate such a matrix?

Answer 1

[Bruno Luong]

http://www.mathworks.com/matlabcentral/fileexchange/17818-all-permutations-of-integers-with-sum-criteria

   K=3
   allVL1(K+1,K,'<=')

ans =

     0     0     0     0
     0     0     0     1
     0     0     1     0
     0     1     0     0
     1     0     0     0
     0     0     0     2
     0     0     1     1
     0     0     2     0
     0     1     0     1
     0     1     1     0
     0     2     0     0
     1     0     0     1
     1     0     1     0
     1     1     0     0
     2     0     0     0
     0     0     0     3
     0     0     1     2
     0     0     2     1
     0     0     3     0
     0     1     0     2
     0     1     1     1
     0     1     2     0
     0     2     0     1
     0     2     1     0
     0     3     0     0
     1     0     0     2
     1     0     1     1
     1     0     2     0
     1     1     0     1
     1     1     1     0
     1     2     0     0
     2     0     0     1
     2     0     1     0
     2     1     0     0
     3     0     0     0

Discard the first row if you don't need all-zero vector.