Mfcttrf

Writing:

n = 3;
Tuples["min", "MAX", n]

I get:

"min", "min", "min", "min", "min", "MAX", "min", "MAX", "min", "min", "MAX", "MAX", "MAX", "min", "min", "MAX", "min","MAX", "MAX", "MAX", "min", "MAX", "MAX", "MAX"

One way to duplicate this function is to fill in this table by columns. In fact, writing:

tuples = ConstantArray[0, 2^n, n];
flag = 1;
For[j = 1, j <= n, j++,
 For[i = 1, i <= 2^n, i++,
 If[flag == 1,
 tuples = ReplacePart[tuples, i, j -> "min"],
 tuples = ReplacePart[tuples, i, j -> "MAX"]
 ];
 If[Mod[i, 2^(n - j)] == 0, flag = flag + 1];
 If[flag > 2, flag = 1]
 ]
 ];
tuples

I get:

"min", "min", "min", "min", "min", "MAX", "min", "MAX", "min", "min", "MAX", "MAX", "MAX", "min", "min", "MAX", "min","MAX", "MAX", "MAX", "min", "MAX", "MAX", "MAX"

Now I would like to duplicate this function by filling in this table by rows. My first approach is the following:

tuples = ConstantArray[0, 2^n, n];
For[i = 1, i <= 2^n, i++,
 For[j = 1, j <= n, j++,
 If[2^(j - 1) <= i <= 2^n/2^j,
 tuples = ReplacePart[tuples, i, j -> "min"],
 tuples = ReplacePart[tuples, i, j -> "MAX"]
 ]
 ]
 ];
tuples

which offers only the first correct column:

"min", "MAX", "MAX", "min", "min", "MAX", "min", "MAX", "MAX", "min", "MAX", "MAX", "MAX", "MAX", "MAX", "MAX", "MAX","MAX", "MAX", "MAX", "MAX", "MAX", "MAX", "MAX"

Any idea to correct this code? Thanks!

Noticing that going row-by-row, we're essentially counting in binary (with "min" as 0 and "MAX" as 1), we see that we can use BitAnd[i - 1, 2^(n - j)] == 0 as condition:

n = 3;

tuples = ConstantArray[0, 2^n, n];
For[i = 1, i <= 2^n, i++, 
 For[j = 1, j <= n, j++, 
 If[BitAnd[i - 1, 2^(n - j)] == 0, 
 tuples = ReplacePart[tuples, i, j -> "min"],
 tuples = ReplacePart[tuples, i, j -> "MAX"]
 ]
 ]
];
tuples
(* "min", "min", "min", "min", "min", "MAX", "min", "MAX", "min", 
 "min", "MAX", "MAX", "MAX", "min", "min", "MAX", "min", "MAX",
 "MAX", "MAX", "min", "MAX", "MAX", "MAX" *)

Or, more concise:

tuples = ConstantArray[0, 2^n, n];
Do[
 tuples = ReplacePart[
 tuples,
 i, j -> If[BitAnd[i - 1, 2^(n - j)] == 0, "min", "MAX"]
 ],
 i, 2^n,
 j, n
 ];
tuples
(* same output *)

The condition works by checking the $(n-j)$-th bit of $i-1$ using BitAnd:

• 
  $n-j$ instead of e.g. $j$ because the zeroth bit is the $n$-th column, and the $n-1$-th bit is the first column


• The $-1$ in $i-1$ is needed since row indices start at $1$, but the counting starts at $0$
  

Note: Since the condition is completely stateless (i.e. there is no flag variable or similar), we can simply swap the iteration indices to change the filling order:

tuples = ConstantArray[0, 2^n, n];
Do[
 tuples = ReplacePart[
 tuples,
 i, j -> If[BitAnd[i - 1, 2^(n - j)] == 0, "min", "MAX"]
 ],
 j, n,
 i, 2^n
 ];
tuples
(* same output *)

Lukes answer is excellent but it can be improved. This version is faster as well as more concise.

n = 3;
tuples = ConstantArray[0, 2^n, n];
Do[
 tuples[[i, j]] = If[BitAnd[i - 1, 2^(n - j)] == 0, "min", "MAX"], 
 j, n, i, 2^n];
tuples // MatrixForm

Your For-loop solution can also be improved by applying the same ideas.

flag = 0;
Do[
 If[flag == 0, tuples[[i, j]] = "min", tuples[[i, j]] = "MAX"]; 
 If[Mod[i, 2^(n - j)] == 0, flag = Mod[flag + 1, 2]],
 j, n, i, 2^n];