How to create a multi-index vectors?

I have to create some vectors containing the values of a d-degree n-dimensional multi-index. A d-degree n-dimensional multi-index is a n-tuple such that .

Just to give an example, if I want a 2-degree 3-dimensional multi-index, I have to built the vectors:

I can to create them for the 2-dimensional case (basicly, I create a matrix and take the upper-right part), but when I move to higher dimensions I have no clue how to go on.

Do you have some suggestion?

Answer 1

[Stephen23]

Start by downloading John D'Errico's excellent partitions function:

https://www.mathworks.com/matlabcentral/fileexchange/12009-partitions-of-an-integer

and then using it like this:

d = 2;
n = 3;
P = partitions(d,1:d,n);
N = size(P,1);
C = cell(1,N);
for k = 1:N
    tmp = repelem(1:d,P(k,:));
    tmp(end+1:n) = 0;
    C{k} = unique(perms(tmp),'rows');
end
Z = vertcat(C{:})

For d=2 and n=3 this gives:

Z =
     0     1     1
     1     0     1
     1     1     0
     0     0     2
     0     2     0
     2     0     0

For d=3 and n=4 this gives:

Z =
     0     1     1     1
     1     0     1     1
     1     1     0     1
     1     1     1     0
     0     0     1     2
     0     0     2     1
     0     1     0     2
     0     1     2     0
     0     2     0     1
     0     2     1     0
     1     0     0     2
     1     0     2     0
     1     2     0     0
     2     0     0     1
     2     0     1     0
     2     1     0     0
     0     0     0     3
     0     0     3     0
     0     3     0     0
     3     0     0     0

You might also be interested to read my answer here:

https://www.mathworks.com/matlabcentral/answers/445026-how-can-i-generate-an-array-of-binary-data-of-this-form