Number of Valid Move Combinations On Chessboard.py

class Solution:
    def countCombinations(self, pieces: List[str], positions: List[List[int]]) -> int:
        board = [[set() for _ in range(8)] for _ in range(8)]
        n = len(pieces)
        for pos in positions:
            pos[0] -= 1
            pos[1] -= 1
        all_time = set(range(1, 8))
        def recur(i):
            if i == n:
                return 1
            ans = 0
            line = [(0, 1), (1, 0), (0, -1), (-1, 0)]
            diag = [(1, 1), (1, -1), (-1, 1), (-1, -1)]
            r, c = positions[i]
            if not board[r][c] & all_time:
                board[r][c] |= all_time
                ans += recur(i + 1)
                board[r][c].clear()
            directions = []
            if pieces[i] in ("queen", "rook"):
                directions.extend(line)
            if pieces[i] in ("queen", "bishop"):
                directions.extend(diag)                
            for dr, dc in directions:
                x, y = r + dr, c + dc
                count = 1
                while 0 <= x < 8 and 0 <= y < 8 and count not in board[x][y]:
                    board[x][y].add(count)
                    count += 1
                    rest = set(range(count, 8))
                    if not board[x][y] & rest:
                        board[x][y] |= rest
                        ans += recur(i + 1)
                        board[x][y] -= rest
                    x += dr
                    y += dc
                count -= 1
                x -= dr
                y -= dc
                while count:
                    board[x][y].remove(count)
                    count -= 1
                    x -= dr
                    y -= dc
            return ans
        return recur(0)