<p>There are <code>n</code> soldiers standing in a line. Each soldier is assigned a <strong>unique</strong> <code>rating</code> value.</p>

<p>You have to form a team of 3 soldiers amongst them under the following rules:</p>

<ul>
	<li>Choose 3 soldiers with index (<code>i</code>, <code>j</code>, <code>k</code>) with rating (<code>rating[i]</code>, <code>rating[j]</code>, <code>rating[k]</code>).</li>
	<li>A team is valid if: (<code>rating[i] &lt; rating[j] &lt; rating[k]</code>) or (<code>rating[i] &gt; rating[j] &gt; rating[k]</code>) where (<code>0 &lt;= i &lt; j &lt; k &lt; n</code>).</li>
</ul>

<p>Return the number of teams you can form given the conditions. (soldiers can be part of multiple teams).</p>

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

<pre>
<strong>Input:</strong> rating = [2,5,3,4,1]
<strong>Output:</strong> 3
<strong>Explanation:</strong> We can form three teams given the conditions. (2,3,4), (5,4,1), (5,3,1). 
</pre>

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

<pre>
<strong>Input:</strong> rating = [2,1,3]
<strong>Output:</strong> 0
<strong>Explanation:</strong> We can&#39;t form any team given the conditions.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> rating = [1,2,3,4]
<strong>Output:</strong> 4
</pre>

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

<ul>
	<li><code>n == rating.length</code></li>
	<li><code>3 &lt;= n &lt;= 1000</code></li>
	<li><code>1 &lt;= rating[i] &lt;= 10<sup>5</sup></code></li>
	<li>All the integers in <code>rating</code> are <strong>unique</strong>.</li>
</ul>

class Fenwick: 
    def __init__(self, values: List[int]):
        n = len(values) + 1

        self.tree = [0] * n 
        self.size = n 

        for i in range(n - 1):
            self.update(i + 1, values[i])
            
    def sum(self, i: int) -> int:
        total = 0 
        
        while i > 0:
            total += self.tree[i]
            i -= (i & -i)

        return total

    def query(self, a: int, b: int) -> int:
        return self.sum(b + 1) - self.sum(a)
        
    def update(self, i: int, d: int):
        while i < self.size:
            self.tree[i] += d 
            i += (i & -i)

    def assign(self, i: int, x: int):
        self.update(i + 1, x - self.query(i, i))

class Solution:
    def numTeams(self, rating: List[int]) -> int:
        n = len(rating)

        highest = max(*rating) + 1

        left = Fenwick([0] * highest)
        left.assign(rating[0], 1)

        leftMultis = [0] * n 
        for i in range(1, n):
            leftMultis[i] = left.query(0, rating[i] - 1)
            left.assign(rating[i], 1)
        
        right = Fenwick([0] * highest)
        right.assign(rating[-1], 1)

        rightMultis = [0] * n
        for i in range(n - 2, -1, -1):
            rightMultis[i] = right.query(rating[i] + 1, highest - 1)
            right.assign(rating[i], 1)

        teams = 0 
        for m in range(n):
            leftSmaller = leftMultis[m]
            leftLarger = m - leftSmaller 

            rightLarger = rightMultis[m]
            rightSmaller = (n - m - 1) - rightLarger 

            teams += leftSmaller * rightLarger
            teams += leftLarger * rightSmaller 

        return teams 
18800