Buggy python submission (verdict: Time Limit Exceeded):
from collections import defaultdict

N = int(input())
S = str(input())

dic = defaultdict(int)

for i in range(2 ** N):
    a = str()
    b = str()
    for j in range(N):
        if (i >> j) & 1:
            a += S[j + N]
        else:
            b += S[j + N]
    
    dic[(a, b)] += 1

count = 0
for i in range(2 ** N):
    a = str()
    b = str()
    for j in range(N):
        if (i >> j) & 1:
            a += S[j]
        else:
            b += S[j]
    count += dic[b[::-1], a[::-1]]

print (count)

Fixed submission:
from collections import defaultdict

N = int(input())
S = str(input())

dic = defaultdict(int)

for i in range(2 ** N):
    a = str()
    b = str()
    for j in range(N):
        if (i >> j) & 1:
            a += S[j + N]
        else:
            b += S[j + N]
    
    dic[(a, b)] += 1

count = 0
for i in range(2 ** N):
    a = str()
    b = str()
    for j in range(N):
        if (i >> j) & 1:
            a += S[j]
        else:
            b += S[j]
    count += dic[b[::-1], a[::-1]]

print (count)

Buggy python submission (verdict: Time Limit Exceeded):
def main():
    N = int(input())
    S = tuple(input())

    def check(red, blue): #赤色、青色がわかるリストが与えられる。
        max_n = len(blue) #青色の個数 = 赤色の後ろにつける個数
        max_r = len(red)
        dp = [[[-1] * (max_n + 1) for _ in range(max_r + 1)] for _ in range(N + 1)]
        
        def calc(i, tmp, n):
            if i == N - 1:
                return 1

            if dp[i - N][tmp][n] != -1:
                return dp[i - N][tmp][n]

            tmp_count = 0
            if n < max_n and S[i] == S[blue[n]]:
                tmp_count += calc(i - 1, tmp, n + 1)
            if tmp < max_r and S[i] == S[red[tmp]]:
                tmp_count += calc(i - 1, tmp + 1, n)
            
            dp[i - N][tmp][n] = tmp_count
            return tmp_count

        return calc(2 * N - 1, 0, 0)
        # for i in range(2 * N - 1, N - 1, -1): #後半部分を後ろから探索
        #     # print (i, n - 1, blue[n - 1])
        #     if n < max_n and S[i] == S[blue[n]]: #青色と一致しているとき
        #         n += 1
        #     else:
        #         if tmp < len(red) and S[i] == S[red[tmp]]:
        #             tmp += 1
        #         else:
        #             return False
        # return True


    count = 0
    for i in range(2 ** N):
        red = []
        blue = []
        # lst = [0] * N #0: 赤色、1: 青色
        for j in range(N):
            if (i >> j) & 1:
                red.append(j)
                # lst[j] = 1
            else:
                blue.append(j)
        
        # print (check(red, blue), red, blue)
        count += check(red, blue)

    print (count)

if __name__ == '__main__':
    main()

Fixed submission:
from collections import defaultdict

N = int(input())
S = str(input())

dic = defaultdict(int)

for i in range(2 ** N):
    a = str()
    b = str()
    for j in range(N):
        if (i >> j) & 1:
            a += S[j + N]
        else:
            b += S[j + N]
    
    dic[(a, b)] += 1

count = 0
for i in range(2 ** N):
    a = str()
    b = str()
    for j in range(N):
        if (i >> j) & 1:
            a += S[j]
        else:
            b += S[j]
    count += dic[b[::-1], a[::-1]]

print (count)

Buggy python submission (verdict: Time Limit Exceeded):
N = int(input())
S = list(input())

def check(red, blue): #赤色、青色がわかるリストが与えられる。
    max_n = len(blue) #青色の個数 = 赤色の後ろにつける個数
    max_r = len(red)
    dp = [[[-1] * (max_n + 1) for _ in range(max_r + 1)] for _ in range(N + 1)]
    
    def calc(red, blue, i, tmp, n):
        if i == N - 1:
            return 1

        if dp[i - N][tmp][n] != -1:
            return dp[i - N][tmp][n]

        tmp_count = 0
        if n < max_n and S[i] == S[blue[n]]:
            tmp_count += calc(red, blue, i - 1, tmp, n + 1)
        if tmp < max_r and S[i] == S[red[tmp]]:
            tmp_count += calc(red, blue, i - 1, tmp + 1, n)
        
        dp[i - N][tmp][n] = tmp_count
        return tmp_count

    return calc(red, blue, 2 * N - 1, 0, 0)
    # for i in range(2 * N - 1, N - 1, -1): #後半部分を後ろから探索
    #     # print (i, n - 1, blue[n - 1])
    #     if n < max_n and S[i] == S[blue[n]]: #青色と一致しているとき
    #         n += 1
    #     else:
    #         if tmp < len(red) and S[i] == S[red[tmp]]:
    #             tmp += 1
    #         else:
    #             return False
    # return True


count = 0
for i in range(2 ** N):
    red = []
    blue = []
    lst = [0] * N #0: 赤色、1: 青色
    for j in range(N):
        if (i >> j) & 1:
            red.append(j)
            lst[j] = 1
        else:
            blue.append(j)
    
    # print (check(red, blue), red, blue)
    count += check(red, blue)

print (count)

Fixed submission:
from collections import defaultdict

N = int(input())
S = str(input())

dic = defaultdict(int)

for i in range(2 ** N):
    a = str()
    b = str()
    for j in range(N):
        if (i >> j) & 1:
            a += S[j + N]
        else:
            b += S[j + N]
    
    dic[(a, b)] += 1

count = 0
for i in range(2 ** N):
    a = str()
    b = str()
    for j in range(N):
        if (i >> j) & 1:
            a += S[j]
        else:
            b += S[j]
    count += dic[b[::-1], a[::-1]]

print (count)

Buggy python submission (verdict: Time Limit Exceeded):
N = int(input())
S = list(input())

def check(red, blue): #赤色、青色がわかるリストが与えられる。
    max_n = len(blue) #青色の個数 = 赤色の後ろにつける個数
    max_r = len(red)
    dp = [[[-1] * (max_n + 1) for _ in range(max_r + 1)] for _ in range(N + 1)]
    
    def calc(red, blue, i, tmp, n):
        if i == N - 1:
            return 1

        if dp[i - N][tmp][n] != -1:
            return dp[i - N][tmp][n]

        tmp_count = 0
        if n < max_n and S[i] == S[blue[n]]:
            tmp_count += calc(red, blue, i - 1, tmp, n + 1)
        if tmp < max_r and S[i] == S[red[tmp]]:
            tmp_count += calc(red, blue, i - 1, tmp + 1, n)
        
        dp[i - N][tmp][n] = tmp_count
        return tmp_count

    return calc(red, blue, 2 * N - 1, 0, 0)
    # for i in range(2 * N - 1, N - 1, -1): #後半部分を後ろから探索
    #     # print (i, n - 1, blue[n - 1])
    #     if n < max_n and S[i] == S[blue[n]]: #青色と一致しているとき
    #         n += 1
    #     else:
    #         if tmp < len(red) and S[i] == S[red[tmp]]:
    #             tmp += 1
    #         else:
    #             return False
    # return True


count = 0
for i in range(2 ** N):
    red = []
    blue = []
    lst = [0] * N #0: 赤色、1: 青色
    for j in range(N):
        if (i >> j) & 1:
            red.append(j)
            lst[j] = 1
        else:
            blue.append(j)
    
    # print (check(red, blue), red, blue)
    count += check(red, blue)

print (count)

Fixed submission:
from collections import defaultdict

N = int(input())
S = str(input())

dic = defaultdict(int)

for i in range(2 ** N):
    a = str()
    b = str()
    for j in range(N):
        if (i >> j) & 1:
            a += S[j + N]
        else:
            b += S[j + N]
    
    dic[(a, b)] += 1

count = 0
for i in range(2 ** N):
    a = str()
    b = str()
    for j in range(N):
        if (i >> j) & 1:
            a += S[j]
        else:
            b += S[j]
    count += dic[b[::-1], a[::-1]]

print (count)