问题

### Instructions: You are an expert Python programmer. You will be given a question (problem specification) and will generate a correct Python program that matches the specification and passes all tests. You will NOT return anything except for the program.
### Question:
You are given an integer n and a 0-indexed integer array sick which is sorted in increasing order.
There are n children standing in a queue with positions 0 to n - 1 assigned to them. The array sick contains the positions of the children who are infected with an infectious disease. An infected child at position i can spread the disease to either of its immediate neighboring children at positions i - 1 and i + 1 if they exist and are currently not infected. At most one child who was previously not infected can get infected with the disease in one second.
It can be shown that after a finite number of seconds, all the children in the queue will get infected with the disease. An infection sequence is the sequential order of positions in which all of the non-infected children get infected with the disease. Return the total number of possible infection sequences.
Since the answer may be large, return it modulo 10^9 + 7.
Note that an infection sequence does not contain positions of children who were already infected with the disease in the beginning.

Example 1:

Input: n = 5, sick = [0,4]
Output: 4
Explanation: Children at positions 1, 2, and 3 are not infected in the beginning. There are 4 possible infection sequences:
- The children at positions 1 and 3 can get infected since their positions are adjacent to the infected children 0 and 4. The child at position 1 gets infected first.
Now, the child at position 2 is adjacent to the child at position 1 who is infected and the child at position 3 is adjacent to the child at position 4 who is infected, hence either of them can get infected. The child at position 2 gets infected.
Finally, the child at position 3 gets infected because it is adjacent to children at positions 2 and 4 who are infected. The infection sequence is [1,2,3].
- The children at positions 1 and 3 can get infected because their positions are adjacent to the infected children 0 and 4. The child at position 1 gets infected first.
Now, the child at position 2 is adjacent to the child at position 1 who is infected and the child at position 3 is adjacent to the child at position 4 who is infected, hence either of them can get infected. The child at position 3 gets infected.
Finally, the child at position 2 gets infected because it is adjacent to children at positions 1 and 3 who are infected. The infection sequence is [1,3,2].
- The infection sequence is [3,1,2]. The order of infection of disease in the children can be seen as: [0,1,2,3,4] => [0,1,2,3,4] => [0,1,2,3,4] => [0,1,2,3,4].
- The infection sequence is [3,2,1]. The order of infection of disease in the children can be seen as: [0,1,2,3,4] => [0,1,2,3,4] => [0,1,2,3,4] => [0,1,2,3,4].

Example 2:

Input: n = 4, sick = [1]
Output: 3
Explanation: Children at positions 0, 2, and 3 are not infected in the beginning. There are 3 possible infection sequences:
- The infection sequence is [0,2,3]. The order of infection of disease in the children can be seen as: [0,1,2,3] => [0,1,2,3] => [0,1,2,3] => [0,1,2,3].
- The infection sequence is [2,0,3]. The order of infection of disease in the children can be seen as: [0,1,2,3] => [0,1,2,3] => [0,1,2,3] => [0,1,2,3].
- The infection sequence is [2,3,0]. The order of infection of disease in the children can be seen as: [0,1,2,3] => [0,1,2,3] => [0,1,2,3] => [0,1,2,3].


Constraints:

2 <= n <= 10^5
1 <= sick.length <= n - 1
0 <= sick[i] <= n - 1
sick is sorted in increasing order.

### Format: You will use the following starter code to write the solution to the problem and enclose your code within delimiters.
```python
class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:

```

### Answer: (use the provided format with backticks)

答案

模型 模型答案 正确答案
金声玉亮2.0 ```python
class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7
sick.sort()
total_ways = 1
prev = -1

for i in range(len(sick)):
if prev == -1:
left_count = sick[i]
else:
left_count = sick[i] - prev - 1

if i == len(sick) - 1:
right_count = n - sick[i] - 1
else:
right_count = sick[i + 1] - sick[i] - 1

total_ways *= self.comb(left_count + right_count, left_count)
total_ways %= MOD
prev = sick[i]

return total_ways

def comb(self, n: int, k: int) -> int:
MOD = 10**9 + 7
if k > n:
return 0
if k > n - k:
k = n - k
comb = 1
for i in range(k):
comb = comb * (n - i) * pow(i + 1, MOD - 2, MOD) % MOD
return comb
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
deepseek-r1

```python
from typing import List

MOD = 10**9 + 7

class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7
max_fact = 10**5 # Precompute up to 1e5 as per problem constraints

# Precompute factorial and inverse factorial modulo MOD
fact = [1] * (max_fact + 1)
for i in range(1, max_fact + 1):
fact[i] = fact[i-1] * i % MOD

inv_fact = [1] * (max_fact + 1)
inv_fact[max_fact] = pow(fact[max_fact], MOD-2, MOD)
for i in range(max_fact - 1, -1, -1):
inv_fact[i] = inv_fact[i + 1] * (i + 1) % MOD

intervals = []

# Interval before the first sick element
first = sick[0]
if first > 0:
m = first # Positions from 0 to first-1
intervals.append((m, 1))

# Intervals between consecutive sick elements
for i in range(len(sick) - 1):
a, b = sick[i], sick[i+1]
m = b - a - 1
if m <= 0:
continue
intervals.append((m, pow(2, m-1, MOD)))

# Interval after the last sick element
last = sick[-1]
if last < n - 1:
m = (n - 1) - last # Positions from last+1 to n-1
intervals.append((m, 1))

product_poss = 1
product_inv_fact = 1
total = 0

for m, poss in intervals:
product_poss = product_poss * poss % MOD
product_inv_fact = product_inv_fact * inv_fact[m] % MOD
total += m

if total == 0:
return 0

result = fact[total] * product_inv_fact % MOD
result = result * product_poss % MOD
return result
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
gpto1 ```python
class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
from math import comb
import sys
import threading
def main():
mod = 10 ** 9 + 7
sick_set = set(sick)
sick = sorted(sick)
positions = []
prev = -1
res = 1
for s in sick:
if s - prev > 1:
k = s - prev -1
res = res * pow(2, k -1, mod) % mod
prev = s
if n - 1 - sick[-1] > 0:
k = n - 1 - sick[-1]
res = res * pow(2, k -1, mod) % mod
print(res)
threading.Thread(target=main).start()
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
gpt-4o-2024-08-06 ```python
class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7

# Calculate the number of uninfected children between each pair of sick children
gaps = []
for i in range(1, len(sick)):
gaps.append(sick[i] - sick[i-1] - 1)

# Add the gaps before the first sick child and after the last sick child
if sick[0] > 0:
gaps.insert(0, sick[0])
if sick[-1] < n - 1:
gaps.append(n - 1 - sick[-1])

# Calculate the number of ways to infect each gap
def factorial(x):
result = 1
for i in range(2, x + 1):
result = (result * i) % MOD
return result

def binomial_coefficient(n, k):
if k > n:
return 0
if k == 0 or k == n:
return 1
num = factorial(n)
denom = (factorial(k) * factorial(n - k)) % MOD
return (num * pow(denom, MOD - 2, MOD)) % MOD

total_ways = 1
for gap in gaps:
if gap > 0:
total_ways = (total_ways * factorial(gap)) % MOD

return total_ways
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
Qwen2-72B-Instruct ```python
from math import factorial

class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7

# Add virtual elements at the beginning and end
sick = [0] + sick + [n]

# Calculate the number of sequences between each pair of sick children
sequences = []
for i in range(1, len(sick)):
gap = sick[i] - sick[i-1] - 1
if gap > 0:
sequences.append(gap)

# Calculate the total number of sequences
total_sequences = factorial(sum(sequences))
for seq in sequences:
total_sequences //= factorial(seq)

# Calculate the number of ways to distribute the gaps among the sick children
distribution = 1
for seq in sequences:
distribution *= pow(2, seq-1, MOD)
distribution %= MOD

# Return the final result modulo 10^9 + 7
return (total_sequences * distribution) % MOD
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
DeepSeek-Coder-V2-Lite-Instruct ```python
class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7
# Create a list to mark infected positions
infected = [False] * n
for pos in sick:
infected[pos] = True

# Calculate the number of ways to infect each child
dp = [0] * n
if not sick:
# If no sick children, all can be infected
for i in range(n):
dp[i] = 1
else:
# Calculate the number of ways to infect each child
for i in range(n):
if not infected[i]:
if i == 0:
dp[i] = dp[i + 1]
elif i == n - 1:
dp[i] = dp[i - 1]
else:
dp[i] = dp[i - 1] + dp[i + 1]

# Calculate the total number of sequences
total_sequences = 1
for i in range(n):
if not infected[i]:
total_sequences = (total_sequences * dp[i]) % MOD

return total_sequences
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
Meta-Llama-3.1-405B-Instruct-Turbo ```python
from typing import List

class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7
res = 1
sick = [-1] + sick + [n]

for i in range(1, len(sick)):
res = (res * self.count(sick[i] - sick[i-1] - 1)) % MOD

return res

def count(self, n: int) -> int:
if n <= 1:
return 1
res = 1
for i in range(1, n+1):
res = (res * i) % (10**9 + 7)
return res
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
Meta-Llama-3.1-70B-Instruct-Turbo ```python
class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7
res = 1
for i in range(len(sick)):
if i == 0:
res = res * max(sick[i] - 0, 1) % MOD
else:
res = res * max(sick[i] - sick[i-1] - 1, 1) % MOD
res = res * max(n - 1 - sick[-1], 1) % MOD
return res
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
baichuan4 ```python
class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7
dp = [0] * (n + 1)
dp[0] = 1

for i in range(1, n + 1):
if i in sick:
dp[i] = dp[i - 1]
else:
dp[i] = (dp[i - 1] * 2) % MOD

return dp[n]
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
doubao ```python
class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7
dp = [0] * n
dp[0] = 1
for i in range(1, n):
if i in sick:
dp[i] = 0
else:
if i - 1 >= 0 and i - 1 not in sick:
dp[i] += dp[i - 1]
if i + 1 < n and i + 1 not in sick:
dp[i] += dp[i + 1]
dp[i] %= MOD
return sum(dp) % MOD
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
gpto1 ```python
class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
from math import comb
import sys
import threading
def main():
mod = 10 ** 9 + 7
sick_set = set(sick)
sick = sorted(sick)
positions = []
prev = -1
res = 1
for s in sick:
if s - prev > 1:
k = s - prev -1
res = res * pow(2, k -1, mod) % mod
prev = s
if n - 1 - sick[-1] > 0:
k = n - 1 - sick[-1]
res = res * pow(2, k -1, mod) % mod
print(res)
threading.Thread(target=main).start()
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
kimi ```python
from typing import List

class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7
dp = [0] * (n + 1)
dp[0] = 1
prev = -1
for i in range(len(sick)):
cur = sick[i]
diff = cur - prev - 1
dp[cur + 1] = (dp[cur] + dp[cur + 1]) % MOD
if diff > 0:
dp[cur + 1] = (dp[cur + 1] * (diff + 1) % MOD + dp[cur + 2]) % MOD
prev = cur
diff = n - 1 - prev
if diff > 0:
dp[-1] = dp[-1] * (diff + 1) % MOD
return dp[-1]
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
minimax ```python
from typing import List

class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7
# sick is sorted in increasing order, so we can find the gaps between infected children
# and calculate the number of ways to infect the children in each gap
sick.append(n) # add the last position to handle the end of the queue
gaps = []
for i in range(len(sick) - 1):
gaps.append(sick[i+1] - sick[i] - 1)

# Calculate the total number of ways to infect the children in the gaps
total_sequences = 1
for gap in gaps:
if gap > 0:
# For each gap, the number of ways to infect is the sum of the previous gap's ways
# and the ways from the current gap. This is because we can infect either end first.
total_sequences *= (gap + 1)
total_sequences %= MOD

return total_sequences

# Example usage:
# sol = Solution()
# print(sol.numberOfSequence(5, [0, 4])) # Output:		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
wenxinyiyan ```python
class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7
dp = [0] * n
sick.append(n) # Add a sentinel value to simplify the logic
prev_sick = -1
total_sequences = 1

for i in range(len(sick)):
if i > 0:
gap = sick[i] - prev_sick - 1

# If there is a gap between sick children, calculate the number of sequences
if gap > 0:
dp[i] = (2 ** gap - 1) % MOD

# Update the total number of sequences
total_sequences = (total_sequences * dp[i]) % MOD

prev_sick = sick[i]

# Handle the last gap (if any)
last_gap = n - 1 - prev_sick
if last_gap > 0:
dp.append((2 ** last_gap - 1) % MOD)
total_sequences = (total_sequences * dp[-1]) % MOD

# Return the total number of sequences modulo MOD
return total_sequences
```

Note: The provided solution uses dynamic programming to calculate the number of possible infection sequences. It iterates over the sick children, calculating the gaps between them and using these gaps to determine the number of sequences possible in each gap. The total number of sequences is then updated by multiplying the sequences in each gap. Finally, the result is returned modulo `10^9 + 7` to satisfy the constraint.

However, please note that the provided solution assumes that the `sick` list is sorted in increasing order, as stated in the problem specification. If the input `sick` list is not sorted, you may need to add a sorting step before proceeding with the calculation.

Additionally, the solution appends a sentinel value `n` to the `sick` list to simplify the logic of handling the last gap. This ensures that the last gap, if any, is also considered in the calculation.		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
zhipu ```python
class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
MOD = 10**9 + 7

# Helper function to calculate factorial % MOD
def factorial(x):
result = 1
for i in range(2, x + 1):
result = result * i % MOD
return result

# Helper function to calculate x^y % MOD
def mod_exp(x, y):
result = 1
while y > 0:
if y % 2 == 1:
result = result * x % MOD
x = x * x % MOD
y //= 2
return result

# Helper function to calculate nCr % MOD
def nCr(n, r):
if r > n:
return 0
num = factorial(n)
denom = factorial(r) * factorial(n - r) % MOD
return num * mod_exp(denom, MOD - 2) % MOD

# Calculate the number of non-infected children
non_infected = n - len(sick)

# If there are no non-infected children, return 0
if non_infected == 0:
return 0

# Calculate the number of gaps between infected children
gaps = []
if sick[0] > 0:
gaps.append(sick[0])
for i in range(1, len(sick)):
if sick[i] - sick[i - 1] > 1:
gaps.append(sick[i] - sick[i - 1] - 1)
if sick[-1] < n - 1:
gaps.append(n - 1 - sick[-1])

# Calculate the number of possible infection sequences
result = 1
for gap in gaps:
result = result * nCr(non_infected, gap) % MOD
non_infected -= gap

return result
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]
deepseek-v3 ```python
MOD = 10**9 + 7

def comb(n, k, factorial, inv_factorial):
if k < 0 or k > n:
return 0
return (factorial[n] * inv_factorial[k] % MOD) * inv_factorial[n - k] % MOD

def precompute_factorials(max_n, MOD):
factorial = [1] * (max_n + 1)
for i in range(1, max_n + 1):
factorial[i] = factorial[i-1] * i % MOD
inv_factorial = [1] * (max_n + 1)
inv_factorial[max_n] = pow(factorial[max_n], MOD-2, MOD)
for i in range(max_n - 1, -1, -1):
inv_factorial[i] = inv_factorial[i+1] * (i+1) % MOD
return factorial, inv_factorial

class Solution:
def numberOfSequence(self, n: int, sick: List[int]) -> int:
if not sick:
return 0
intervals = []
prev = -1
for s in sick:
if s - prev > 1:
intervals.append((prev + 1, s - 1))
prev = s
if prev < n - 1:
intervals.append((prev + 1, n - 1))
total = 1
max_len = max([end - start + 1 for start, end in intervals] + [0])
factorial, inv_factorial = precompute_factorials(max_len, MOD)
for start, end in intervals:
length = end - start + 1
if length == 0:
continue
total = total * comb(length, length, factorial, inv_factorial) % MOD
return total
```		 [{"input": "5\n[0, 4]", "output": "4", "testtype": "functional"}, {"input": "4\n[1]", "output": "3", "testtype": "functional"}]