Got it!
// Empty
static final int MT = -1;
static int kRooksPermutations(int[] Q, int col, int rooksInHand) {
// Are we at the last col?
if (col >= Q.length) {
// If we've placed K rooks then its a good'n.
return rooksInHand == 0 ? 1 : 0;
}
// Count at this level starts at 0
int count = 0;
// Have we run out of rooks?
if (rooksInHand > 0) {
// No! Try putting one in each row in this column.
for (int row = 0; row < Q.length; row++) {
// Can a rook be placed here?
if (safeToAdd(Q, row, col)) {
// Mark this spot occupied.
Q[col] = row;
// Recurse to the next column with one less rook.
count += kRooksPermutations(Q, col + 1, rooksInHand - 1);
// No longer occupied.
Q[col] = MT;
}
}
}
// Also try NOT putting a rook in this column.
count += kRooksPermutations(Q, col + 1, rooksInHand);
return count;
}
static boolean safeToAdd(int[] Q, int row, int col) {
// Unoccupied!
if (Q[col] != MT) {
return false;
}
// Do any columns have a rook in this row?
// Could probably stop at col here rather than Q.length
for (int c = 0; c < Q.length; c++) {
if (Q[c] == row) {
// Yes!
return false;
}
}
// All clear.
return true;
}
// Main entry - Build the array and start it all going.
private static void kRooksPermutations(int N, int K) {
// One for each column of the board.
// Contains the row number in which a rook is placed or -1 (MT) if the column is empty.
final int[] Q = new int[N];
// Start all empty.
Arrays.fill(Q, MT);
// Start at column 0 with no rooks placed.
int perms = kRooksPermutations(Q, 0, K);
// Print it.
System.out.println("Perms for N = " + N + " K = " + K + " = " + perms);
}
public static void main(String[] args) {
kRooksPermutations(8, 1);
kRooksPermutations(8, 2);
kRooksPermutations(8, 3);
kRooksPermutations(8, 4);
kRooksPermutations(8, 5);
kRooksPermutations(8, 6);
kRooksPermutations(8, 7);
kRooksPermutations(8, 8);
}
Prints:
Perms for N = 8 K = 1 = 64
Perms for N = 8 K = 2 = 1568
Perms for N = 8 K = 3 = 18816
Perms for N = 8 K = 4 = 117600
Perms for N = 8 K = 5 = 376320
Perms for N = 8 K = 6 = 564480
Perms for N = 8 K = 7 = 322560
Perms for N = 8 K = 8 = 40320