<p>You are given an integer <code>n</code> and an array <code>sick</code> sorted in increasing order, representing positions of infected people in a line of <code>n</code> people.</p>

<p>At each step, <strong>one </strong>uninfected person <strong>adjacent</strong> to an infected person gets infected. This process continues until everyone is infected.</p>

<p>An <strong>infection sequence</strong> is the order in which uninfected people become infected, excluding those initially infected.</p>

<p>Return the number of different infection sequences possible, modulo <code>10<sup>9</sup>+7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 5, sick = [0,4]</span></p>

<p><strong>Output:</strong> <span class="example-io">4</span></p>

<p><strong>Explanation:</strong></p>

<p>There is a total of 6 different sequences overall.</p>

<ul>
	<li>Valid infection sequences are <code>[1,2,3]</code>, <code>[1,3,2]</code>, <code>[3,2,1]</code> and <code>[3,1,2]</code>.</li>
	<li><code>[2,3,1]</code> and <code>[2,1,3]</code> are not valid infection sequences because the person at index 2 cannot be infected at the first step.</li>
</ul>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 4, sick = [1]</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<p>There is a total of 6 different sequences overall.</p>

<ul>
	<li>Valid infection sequences are <code>[0,2,3]</code>, <code>[2,0,3]</code> and <code>[2,3,0]</code>.</li>
	<li><code>[3,2,0]</code>, <code>[3,0,2]</code>, and <code>[0,3,2]</code> are not valid infection sequences because the infection starts at the person at index 1, then the order of infection is 2, then 3, and hence 3 cannot be infected earlier than 2.</li>
</ul>
</div>

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

<ul>
	<li><code>2 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= sick.length &lt;= n - 1</code></li>
	<li><code>0 &lt;= sick[i] &lt;= n - 1</code></li>
	<li><code>sick</code> is sorted in increasing order.</li>
</ul>

MOD = 1_000_000_007
MX = 100_000

# 组合数模板
fac = [0] * MX
fac[0] = 1
for i in range(1, MX):
    fac[i] = fac[i - 1] * i % MOD

inv_fac = [0] * MX
inv_fac[MX - 1] = pow(fac[MX - 1], -1, MOD)
for i in range(MX - 1, 0, -1):
    inv_fac[i - 1] = inv_fac[i] * i % MOD

def comb(n: int, k: int) -> int:
    return fac[n] * inv_fac[k] % MOD * inv_fac[n - k] % MOD

class Solution:
    def numberOfSequence(self, n: int, a: List[int]) -> int:
        m = len(a)
        total = n - m
        ans = comb(total, a[0]) * comb(total - a[0], n - a[-1] - 1) % MOD
        total -= a[0] + n - a[-1] - 1
        e = 0
        for p, q in pairwise(a):
            k = q - p - 1
            if k:
                e += k - 1
                ans = ans * comb(total, k) % MOD
                total -= k
        return ans * pow(2, e, MOD) % MOD
#https://leetcode.cn/problems/count-the-number-of-infection-sequences/solutions/2551734/zu-he-shu-xue-ti-by-endlesscheng-5fjp/

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