<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 = int(1e9+7)
mod_fac = cache(lambda n: (mod_fac(n-1)*n) % mod if n else 1)
mod_inv = cache(lambda a: pow(a, mod - 2, mod))
class Solution:
    def numberOfSequence(self, n: int, sick: List[int]) -> int:
        disperse = lambda x, y: mod_fac(y)*(mod_inv(mod_fac(x)*mod_fac(y-x)))
        sick = set(sick)
        #Total steps
        y = n-len(sick)
        #Ways to layer steps
        res = 1
        l = -1
        for r in range(n):
            if r not in sick: continue
            x = r-1-l
            #Skip empty spans
            if not x:
                l = r
                continue
            #c: combinations for this span
            if l == -1:
                c = 1
            else:
                c = pow(2, x-1, mod)
            #d: ways to disperse these combinations
            # into y remaining steps
            d = disperse(x, y)
            res*=c*d
            res%=mod
            y-=x
            l = r
        # print(sorted({randint(0, int(1e5)) for _ in range(10000)}))
        return res
859