Duplicate Tuples

Question

Writing:

Tuples[{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}]

I get:

{{1, 4, 7}, {1, 4, 8}, {1, 4, 9}, {1, 5, 7}, {1, 5, 8}, {1, 5, 9}, {1,6, 7},
 {1, 6, 8}, {1, 6, 9}, {2, 4, 7}, {2, 4, 8}, {2, 4, 9}, {2, 5, 7}, {2,5, 8},
 {2, 5, 9}, {2, 6, 7}, {2, 6, 8}, {2, 6, 9}, {3, 4, 7}, {3, 4, 8}, {3,4, 9},
 {3, 5, 7}, {3, 5, 8}, {3, 5, 9}, {3, 6, 7}, {3, 6, 8}, {3, 6, 9}}

Similarly, by defining the function:

tuples[matrix_] := Module[{A, i, h, j, k, m, n},
  {m, n} = {Length[matrix], Length[Transpose[matrix]]};
  A = ConstantArray[0, {n^m, m}];
  h = 0; k = 1;
  For[j = 1, j <= m, j++,
      h = h + 1;
      For[i = 1, i <= n^m, i++,
          A = ReplacePart[A, {i, j} -> matrix[[h]][[k]]];
          If[Mod[i, n^m/n^h] == 0, k = k + 1];
          If[k > n, k = 1]
] ]; Return[A]]

and writing:

tuples[{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}]

I get:

{{1, 4, 7}, {1, 4, 8}, {1, 4, 9}, {1, 5, 7}, {1, 5, 8}, {1, 5, 9}, {1,6, 7},
 {1, 6, 8}, {1, 6, 9}, {2, 4, 7}, {2, 4, 8}, {2, 4, 9}, {2, 5, 7}, {2,5, 8},
 {2, 5, 9}, {2, 6, 7}, {2, 6, 8}, {2, 6, 9}, {3, 4, 7}, {3, 4, 8}, {3,4, 9},
 {3, 5, 7}, {3, 5, 8}, {3, 5, 9}, {3, 6, 7}, {3, 6, 8}, {3, 6, 9}}

But if I write now:

Tuples[{{1, 2, 3, 4}, {5, 6}, {7, 8, 9}}]

I get:

{{1, 5, 7}, {1, 5, 8}, {1, 5, 9}, {1, 6, 7}, {1, 6, 8}, {1, 6, 9},
 {2, 5, 7}, {2, 5, 8}, {2, 5, 9}, {2, 6, 7}, {2, 6, 8}, {2, 6, 9},
 {3, 5, 7}, {3, 5, 8}, {3, 5, 9}, {3, 6, 7}, {3, 6, 8}, {3, 6, 9},
 {4, 5, 7}, {4, 5, 8}, {4, 5, 9}, {4, 6, 7}, {4, 6, 8}, {4, 6, 9}}

and defining the function:

tuples[matrix_] := Module[{A, i, h, j, k, m, n},
  m = Length[matrix];
  n = 1; For[i = 1, i <= m, i++, n = n Length[matrix[[i]]]];
  A = ConstantArray[0, {n, m}];
  h = 0; k = 1;
  For[j = 1, j <= m, j++,
      h = h + 1;
      For[i = 1, i <= n, i++,
          A = ReplacePart[A, {i, j} -> matrix[[h]][[k]]];
          If[Mod[i, m / h] == 0, k = k + 1];
          If[k > Length[matrix[[j]]], k = 1]
] ]; Return[A]]

and writing:

tuples[{{1, 2, 3, 4}, {5, 6}, {7, 8, 9}}]

I get:

{{1, 5, 7}, {1, 5, 8}, {1, 5, 9}, {2, 6, 7}, {2, 6, 8}, {2, 6, 9},
 {3,5, 7}, {3, 5, 8}, {3, 5, 9}, {4, 6, 7}, {4, 6, 8}, {4, 6, 9}, 
{1, 5, 7}, {1, 5, 8}, {1, 5, 9}, {2, 6, 7}, {2, 6, 8}, {2, 6, 9}, 
{3, 5,7}, {3, 5, 8}, {3, 5, 9}, {4, 6, 7}, {4, 6, 8}, {4, 6, 9}}

which does not match with the output of MMA.

Most likely the problem lies in mod(...) but I just can not figure out how to correct it.

Answer 1

Rather than use For loops, I suggest a route through Outer:

Clear[tuples]
tuples[list_] := 
 Outer[List, Sequence @@ list] // 
  ArrayReshape[#, Dimensions[#] /. {a__, b_} :> {Times[a], b}] &

Testing this:

data = {{1, 2, 3, 4}, {5, 6}, {7, 8, 9}};
tuples[data] == Tuples[data] 

(* Out: True *)