output_description: Print the answer.
  
  * * *
input_description: Input is given from Standard Input in the following format:
  
      
      
      N M
      A_1 A_2 ... A_M
attempt: def cmb(n, r, mod):  # コンビネーションの高速計算
      if r < 0 or r > n:
          return 0
      r = min(r, n - r)
      return g1[n] * g2[r] * g2[n - r] % mod
  
  
  mod = 10**9 + 7  # 出力の制限
  N = 10**5
  g1 = [1] * (N + 1)  # 元テーブル
  g2 = [1] * (N + 1)  # 逆元テーブル
  inverse = [1] * (N + 1)  # 逆元テーブル計算用テーブル
  
  for i in range(2, N + 1):
      g1[i] = (g1[i - 1] * i) % mod
      inverse[i] = (-inverse[mod % i] * (mod // i)) % mod
      g2[i] = (g2[i - 1] * inverse[i]) % mod
  inverse[0] = 0
  
  N, M = map(int, input().split())
  A = [0] + list(map(int, input().split()))
  start = time.time()
  dp = [[0 for i in range(2**N)] for j in range(M + 1)]
  
  dp[0][0] = g1[2**N - 1]
  
  for i in range(1, M + 1):
      for j in range(2**N):
          dp[i][j] = dp[i - 1][j]
          for k in range(N):
              if j >> k & 1 == 0 and 2**N - A[i] - j >= 2**k - 1:
                  dp[i][j] = (
                      dp[i][j]
                      + (
                          (-1)
                          * (
                              (g1[2**N - A[i] - j] * g2[2**N - A[i] + 1 - (j + 2**k)])
                              % mod
                          )
                          * ((dp[i - 1][j + 2**k] + g1[2**N - 1 - j - 2**k]) * 2**k)
                          % mod
                      )
                      % mod
                  ) % mod
  
  # for i in range(M+1):
  # print(dp[i])
  print((dp[M][0] * pow(2, N, mod)) % mod)
  
problem_description: Statement
  
  There are 2^N players, numbered 1, 2, ..., 2^N. They decided to hold a
  tournament.
  
  The tournament proceeds as follows:
  
    * Choose a permutation of 1, 2, ..., 2^N: p_1, p_2, ..., p_{2^N}.
    * The players stand in a row in the order of Player p_1, Player p_2, ..., Player p_{2^N}.
    * Repeat the following until there is only one player remaining in the row:
      * Play the following matches: the first player in the row versus the second player in the row, the third player versus the fourth player, and so on. The players who lose leave the row. The players who win stand in a row again, preserving the relative order of the players.
    * The last player who remains in the row is the champion.
  
  It is known that, the result of the match between two players can be written
  as follows, using M integers A_1, A_2, ..., A_M given as input:
  
    * When y = A_i for some i, the winner of the match between Player 1 and Player y (2 \leq y \leq 2^N) will be Player y.
    * When y \neq A_i for every i, the winner of the match between Player 1 and Player y (2 \leq y \leq 2^N) will be Player 1.
    * When 2 \leq x < y \leq 2^N, the winner of the match between Player x and Player y will be Player x.
  
  The champion of this tournament depends only on the permutation p_1, p_2, ...,
  p_{2^N} chosen at the beginning. Find the number of permutation p_1, p_2, ...,
  p_{2^N} chosen at the beginning of the tournament that would result in Player
  1 becoming the champion, modulo 10^9 + 7.
samples: [{"input": "2 1\n    3", "output": "8\n    \n\nExamples of p that satisfy the condition are: [1, 4, 2, 3] and [3, 2, 1, 4].\nExamples of p that do not satisfy the condition are: [1, 2, 3, 4] and [1, 3,\n2, 4].\n\n* * *"}, {"input": "4 3\n    2 4 6", "output": "0\n    \n\n* * *"}, {"input": "3 0", "output": "40320\n    \n\n* * *"}, {"input": "3 3\n    3 4 7", "output": "2688\n    \n\n* * *"}, {"input": "16 16\n    5489 5490 5491 5492 5493 5494 5495 5497 18993 18995 18997 18999 19000 19001 19002 19003", "output": "816646464"}]