How to create all possible matrices of zeros and ones without having more than one '1' in the same column?

I want to create all possible matrices of zeros and ones without having more than one '1' in the same column and then save each combination individually

For Example n=number of rows and m= number of column. So the possible combination number is C=n^m. Assuming n=2 and m=3 so C=8. And I need the output to be in a matrix form for each combination as shown below.

X1 = [1 1 1

0 0 0]

X2 = [0 0 0

1 1 1]

X3 = [1 0 0

0 1 1]

X4 = [0 1 1

1 0 0]

X5 = [1 1 0

0 0 1]

X6 = [0 0 1

1 1 0]

X7 = [1 0 1

0 1 0]

X8 = [0 1 0

1 0 1]

[Steven Lord]

You're missing several possibilities. One such example is

z = zeros(2, 3)

z = 2×3

0 0 0 0 0 0

This has no more than one 1 in the same column. Is this a valid matrix you want your function to generate?

[Bruno Luong]

The problem suggested by @Steven Lord is almost the same problem for exactly one 1 by column:

Simply generate for all possible (m+1) x n matrix with exactly one 1 by column; then thow away one row, as the last row.

m = 2;
n = 3;

c = cell(1,n);
[c{:}] = ndgrid(1:m+1); % change here
i = reshape(cat(n+1,c{:}),[],n);
[j,k] = meshgrid(1:n,1:size(i,1));
X = accumarray([i(:) j(:) k(:)],1);
X(end,:,:) = []; % new here

X

X = 
X(:,:,1) =

     1     1     1
     0     0     0


X(:,:,2) =

     0     1     1
     1     0     0


X(:,:,3) =

     0     1     1
     0     0     0


X(:,:,4) =

     1     0     1
     0     1     0


X(:,:,5) =

     0     0     1
     1     1     0


X(:,:,6) =

     0     0     1
     0     1     0


X(:,:,7) =

     1     0     1
     0     0     0


X(:,:,8) =

     0     0     1
     1     0     0


X(:,:,9) =

     0     0     1
     0     0     0


X(:,:,10) =

     1     1     0
     0     0     1


X(:,:,11) =

     0     1     0
     1     0     1


X(:,:,12) =

     0     1     0
     0     0     1


X(:,:,13) =

     1     0     0
     0     1     1


X(:,:,14) =

     0     0     0
     1     1     1


X(:,:,15) =

     0     0     0
     0     1     1


X(:,:,16) =

     1     0     0
     0     0     1


X(:,:,17) =

     0     0     0
     1     0     1


X(:,:,18) =

     0     0     0
     0     0     1


X(:,:,19) =

     1     1     0
     0     0     0


X(:,:,20) =

     0     1     0
     1     0     0


X(:,:,21) =

     0     1     0
     0     0     0


X(:,:,22) =

     1     0     0
     0     1     0


X(:,:,23) =

     0     0     0
     1     1     0


X(:,:,24) =

     0     0     0
     0     1     0


X(:,:,25) =

     1     0     0
     0     0     0


X(:,:,26) =

     0     0     0
     1     0     0


X(:,:,27) =

     0     0     0
     0     0     0

Answer 1

[Bruno Luong]

m = 2;
n = 3;

c = cell(1,n);
[c{:}] = ndgrid(1:m);
i = reshape(cat(n+1,c{:}),[],n);
[j,k] = meshgrid(1:n,1:size(i,1));
X = accumarray([i(:) j(:) k(:)],1)

X = 
X(:,:,1) =

     1     1     1
     0     0     0


X(:,:,2) =

     0     1     1
     1     0     0


X(:,:,3) =

     1     0     1
     0     1     0


X(:,:,4) =

     0     0     1
     1     1     0


X(:,:,5) =

     1     1     0
     0     0     1


X(:,:,6) =

     0     1     0
     1     0     1


X(:,:,7) =

     1     0     0
     0     1     1


X(:,:,8) =

     0     0     0
     1     1     1

[reem alhadrami]

Thank you

Answer 2

[Jan]

n = 3;
m = 4;
c = n^m;
X = cell(1, c);    % List of outputs
v = n .^ (1-m:0);  % Calculate expensive power operation once
for ic = 0:c - 1
  index = rem(floor(ic .* v), n);  % Column indices of 1s
  XX    = zeros(n, m);
  XX(sub2ind([n, m], index + 1, 1:m)) = 1;
  X{ic + 1} = XX;
end
X{1}, X{2}, X{end-1}, X{end}

ans = 3×4

1 1 1 1 0 0 0 0 0 0 0 0

ans = 3×4

1 1 1 0 0 0 0 1 0 0 0 0

ans = 3×4

0 0 0 0 0 0 0 1 1 1 1 0

ans = 3×4

0 0 0 0 0 0 0 0 1 1 1 1

This chooses m numbers from 1:n as column index of 1s in a nxm matrix of 0s.

[reem alhadrami]

Thank you

[Bruno Luong]

v = n .^ (1-m:0);  % Calculate expensive power operation once
...
    index = rem(floor(ic .* v), n);  % Column indices of 1

@Jan it looks in the border side effect of numerical error to me, do you make any analysis to make sure such formula can be used safely?

[Jan]

@Bruno Luong: You are right - this is instable. I've taken it from dec2bin (it was an m-file in former Matlab versions), but hier n=2 is less susceptible to rounding artifacts.

This is better to produce the column indices:

function I = PermRepOrder(n, m)
a = 1;
b = n^m;
I = zeros(b, m);
v = (1:n).';
for k = 1:m
    b = b / n;
    I(:, k) = repmat(repelem(v, b, 1), a, 1);
    a = a * n;      
end
end

[Bruno Luong]

@Jan I like it, thanks.