How to obtain all permutations of multiple cell arrays?

I have a few cell arrays of character vectors - let's say 3 in this case - A, B, C.

A={'a1'; 'a2'; 'a3'};
B={'b1'; 'b2'};
C={'c1'; 'c2'; 'c3'; 'c4'};

I wish to find all permutations of elements of A, B, and C. That is, one element from each. Such as:

{'a1', 'b1', 'c1'; 'a1', 'c1', 'b1'; 'b1', 'a1', 'c1'; 'a1', 'b1', 'c2'; .... }

Please note that I am not looking for only combinations - I believe that would have been easily addressed by:

X=combinations(A, B, C);

I am also not looking for permutations that would produce combinations/permutations of elements from within one of the cell arrays above - so that I couldn't just do:

X=vertcat(A,B,C);
perms(X);

Is there an efficient way to do this without all sorts of loops?

Answer 1

[Matt J]

This is better because the for loop is only 6 iterations long.

A={'a1'; 'a2'; 'a3'};
B={'b1'; 'b2'};
C={'c1'; 'c2'; 'c3'; 'c4'};

p=perms(1:3);
c= combinations(C,B,A);

N=height(p)

N = 6

Z=cell(N,1);
for i=1:N
 Z{i}=c(:,p(i,:));
end

Z=vertcat(Z{:})

Z = 144x3 table
      A         B         C   
    ______    ______    ______

    {'a1'}    {'b1'}    {'c1'}
    {'a2'}    {'b1'}    {'c1'}
    {'a3'}    {'b1'}    {'c1'}
    {'a1'}    {'b2'}    {'c1'}
    {'a2'}    {'b2'}    {'c1'}
    {'a3'}    {'b2'}    {'c1'}
    {'a1'}    {'b1'}    {'c2'}
    {'a2'}    {'b1'}    {'c2'}
    {'a3'}    {'b1'}    {'c2'}
    {'a1'}    {'b2'}    {'c2'}
    {'a2'}    {'b2'}    {'c2'}
    {'a3'}    {'b2'}    {'c2'}
    {'a1'}    {'b1'}    {'c3'}
    {'a2'}    {'b1'}    {'c3'}
    {'a3'}    {'b1'}    {'c3'}
    {'a1'}    {'b2'}    {'c3'}

Answer 2

[AR]

I suppose I could do:

Z = combinations(A, B, C);
for i=1:size(Z,1)
    Y{i,1} = perms(Z{i, :});
end
X = vertcat(Y{:});

I'm just wondering if there's a better way to do it, or if anyone can spot a reason why the above might fail (typically some dimension mismatch).

Answer 3

[Matt J]

Using this FEX download,

https://www.mathworks.com/matlabcentral/fileexchange/115340-block-transposition-permutation-and-reshaping-of-arrays

c= table2cell(combinations(C,B,A));

result =  blkColon(  c(:,perms(1:3)')  , [1,3])

result =

144×3 cell array

{'a1'} {'b1'} {'c1'}

{'a2'} {'b1'} {'c1'}

{'a3'} {'b1'} {'c1'}

{'a1'} {'b2'} {'c1'}

{'a2'} {'b2'} {'c1'}

{'a3'} {'b2'} {'c1'}

{'a1'} {'b1'} {'c2'}

{'a2'} {'b1'} {'c2'}

{'a3'} {'b1'} {'c2'}

{'a1'} {'b2'} {'c2'}

{'a2'} {'b2'} {'c2'}

{'a3'} {'b2'} {'c2'}

{'a1'} {'b1'} {'c3'}

...