<p>There is an <code>8 x 8</code> chessboard containing <code>n</code> pieces (rooks, queens, or bishops). You are given a string array <code>pieces</code> of length <code>n</code>, where <code>pieces[i]</code> describes the type (rook, queen, or bishop) of the <code>i<sup>th</sup></code> piece. In addition, you are given a 2D integer array <code>positions</code> also of length <code>n</code>, where <code>positions[i] = [r<sub>i</sub>, c<sub>i</sub>]</code> indicates that the <code>i<sup>th</sup></code> piece is currently at the <strong>1-based</strong> coordinate <code>(r<sub>i</sub>, c<sub>i</sub>)</code> on the chessboard.</p>

<p>When making a <strong>move</strong> for a piece, you choose a <strong>destination</strong> square that the piece will travel toward and stop on.</p>

<ul>
	<li>A rook can only travel <strong>horizontally or vertically</strong> from <code>(r, c)</code> to the direction of <code>(r+1, c)</code>, <code>(r-1, c)</code>, <code>(r, c+1)</code>, or <code>(r, c-1)</code>.</li>
	<li>A queen can only travel <strong>horizontally, vertically, or diagonally</strong> from <code>(r, c)</code> to the direction of <code>(r+1, c)</code>, <code>(r-1, c)</code>, <code>(r, c+1)</code>, <code>(r, c-1)</code>, <code>(r+1, c+1)</code>, <code>(r+1, c-1)</code>, <code>(r-1, c+1)</code>, <code>(r-1, c-1)</code>.</li>
	<li>A bishop can only travel <strong>diagonally</strong> from <code>(r, c)</code> to the direction of <code>(r+1, c+1)</code>, <code>(r+1, c-1)</code>, <code>(r-1, c+1)</code>, <code>(r-1, c-1)</code>.</li>
</ul>

<p>You must make a <strong>move</strong> for every piece on the board simultaneously. A <strong>move combination</strong> consists of all the <strong>moves</strong> performed on all the given pieces. Every second, each piece will instantaneously travel <strong>one square</strong> towards their destination if they are not already at it. All pieces start traveling at the <code>0<sup>th</sup></code> second. A move combination is <strong>invalid</strong> if, at a given time, <strong>two or more</strong> pieces occupy the same square.</p>

<p>Return <em>the number of <strong>valid</strong> move combinations</em>​​​​​.</p>

<p><strong>Notes:</strong></p>

<ul>
	<li><strong>No two pieces</strong> will start in the<strong> same</strong> square.</li>
	<li>You may choose the square a piece is already on as its <strong>destination</strong>.</li>
	<li>If two pieces are <strong>directly adjacent</strong> to each other, it is valid for them to <strong>move past each other</strong> and swap positions in one second.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/09/23/a1.png" style="width: 215px; height: 215px;" />
<pre>
<strong>Input:</strong> pieces = [&quot;rook&quot;], positions = [[1,1]]
<strong>Output:</strong> 15
<strong>Explanation:</strong> The image above shows the possible squares the piece can move to.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/09/23/a2.png" style="width: 215px; height: 215px;" />
<pre>
<strong>Input:</strong> pieces = [&quot;queen&quot;], positions = [[1,1]]
<strong>Output:</strong> 22
<strong>Explanation:</strong> The image above shows the possible squares the piece can move to.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/09/23/a3.png" style="width: 214px; height: 215px;" />
<pre>
<strong>Input:</strong> pieces = [&quot;bishop&quot;], positions = [[4,3]]
<strong>Output:</strong> 12
<strong>Explanation:</strong> The image above shows the possible squares the piece can move to.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>n == pieces.length </code></li>
	<li><code>n == positions.length</code></li>
	<li><code>1 &lt;= n &lt;= 4</code></li>
	<li><code>pieces</code> only contains the strings <code>&quot;rook&quot;</code>, <code>&quot;queen&quot;</code>, and <code>&quot;bishop&quot;</code>.</li>
	<li>There will be at most one queen on the chessboard.</li>
	<li><code>1 &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 8</code></li>
	<li>Each <code>positions[i]</code> is distinct.</li>
</ul>

class Solution:
    def countCombinations(self, pieces: List[str], positions: List[List[int]]) -> int:
        positions = [tuple(x) for x in positions]

        def dfs(pos, dirs, stop, seen):   
            if stop == 0: return
            seen.add(tuple(pos))
            for active in range(1<<len(dirs)):
                if stop & active != active: continue
                new_pos = list(pos)
                new_mask = stop ^ active

                for i in range(len(new_pos)):
                    new_pos[i] = (new_pos[i][0] + dirs[i][0]*((new_mask>>i)&1), new_pos[i][1] + dirs[i][1]*((new_mask>>i)&1))

                if len(Counter(new_pos)) < len(dirs): continue
                all_c = list(chain(*new_pos))
                if min(all_c) <= 0 or max(all_c) > 8: continue
                dfs(new_pos, dirs, new_mask, seen)

        moves = {}
        moves["rook"] = ((1, 0), (-1, 0), (0, 1), (0, -1))
        moves["bishop"] = ((1, 1), (1, -1), (-1, 1), (-1, -1))
        moves["queen"] = ((1, 0), (-1, 0), (0, 1), (0, -1), (1, 1), (1, -1), (-1, 1), (-1, -1))
        seen = set()
        for dirs in product(*(moves[i] for i in pieces)):
            dfs(positions, dirs, (1<<len(pieces)) - 1, seen)

        return len(seen)
37500