D for loops concatenated

I would like to generalize the following codes for dimension D=2:

D=2;
dim=factorial(P-1+D)/(factorial(P-1)*factorial(D));   
indexes=zeros(dim,D);
s=0;
for i=1:P
  for j=1:i
      s=s+1;
      indexes(s,:)=[i-j,j-1];
   end
end

dimension D=3:

D=3;
dim=factorial(P-1+D)/(factorial(P-1)*factorial(D)); 
indexes=zeros(dim,D);
s=1;
for i=1:P
    for j=1:i
        for k=1:j
            indexes(s,:)=[i-j,j-k,k-1];
            s=s+1;
        end
    end
end

P can be an arbitrary integer (e.g. P=4).

In other words I would like to create D for loops in such a way that the first loop goes from 1 to P (i=1:P) the second loop from 1 to the previous one (j=1:i) the third one from 1 to the second one (h=1:j) etc.

Thank you in advance.

Answer 1

[Jan]

D       = 3;
dim     = factorial(P-1+D)/(factorial(P-1)*factorial(D)); 
indexes = zeros(dim,D);
v       = ones(1, dim);
for s = 1:dim    
  indexes(s, 1:D-1) = diff(v);
  indexes(s, D)     = v(D) - 1;

    % Update index vector:
    found = false;
    for k = D:-1:1
      v(k) = v(k) + 1;

      if k > 1
        if v(k) <= v(k - 1)
          break;  % Exit for k loop
        end
        v(k) = 1;
      end
    end
  end

Please test and adjust this to your needs. The idea gets clear: Use one loop and a vector of indices, which are incremented according to the rules in each iteration.

Answer 2

[Luca Fenzi]

Thank you very much for your answer I adjust it in the following way

 P=5;
D =3;
dim     = nchoosek(P-1+D,D); 
indexes = zeros(dim,D);
v       = ones(1,D);
for s = 1:dim    
  indexes(s, 1:D-1) = -diff(v);
  indexes(s, D)     = v(D) - 1;
    % Update index vector:
    for k = D:-1:1
      v(k) = v(k) + 1;
      if k > 1
        if v(k) <= v(k - 1);
          break;  % Exit for k loop
        end
        v(k) = 1;
      end
    end
end

Here I add also an other code for general for-loops concatenated

N=P^D;
indexes=zeros(P,D);
row=zeros(1,D);
for s=1:N
  indexes(s,:)=row;

    %%Updating
    %the alghorithm is counting how many indexes arrived to P-1 (Maximum
    %degree)
    h=0;
    while (h<=D-2)&&(row(h+1)==P-1)
        h=h+1;
    end

    if h>0
        row(1:h)=zeros(1,h);   %Put zero all the values that were P-1
        row(h+1)=row(h+1)+1;   % add one to the next index.
    end
    row(1)=mod(s,P);
end

(Please feel free to comment and improve my codes so I can learn something new =) )