<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>

def count(T, l, r):
    if l > r:
        return 0
    if l:
        return count(T, 0, r) - count(T, 0, l - 1)
    res = 0
    while r >= 0:
        res += T[r]
        r = (r & (r + 1)) - 1
    return res


def update(T, n, i, amt):
    while i < n:
        T[i] += amt
        i = i | (i + 1)


class Solution:
    def numTeams(self, rating: List[int]) -> int:
        # low mem
        n = len(rating)
        tot = 0
        for i in range(n):
            left = right = 0
            for j in range(i):
                if rating[j] < rating[i]:
                    left += 1
            for j in range(i + 1, n):
                if rating[j] < rating[i]:
                    right += 1
            tot += left * (n - i - 1 - right) + (i - left) * right

        return tot

    def numTeams(self, rating: List[int]) -> int:
        # low time
        n = max(rating) + 1
        left = [0] * n
        right = [0] * n
        for rat in rating:
            update(right, n, rat, 1)
        tot = 0
        for rat in rating:
            l = count(left, 0, rat)
            if l:
                r = count(right, rat + 1, n - 1)
                tot += l * r
            l = count(left, rat, n - 1)
            if l:
                r = count(right, 0, rat - 1)
                tot += l * r
            update(left, n, rat, 1)
            update(right, n, rat, -1)
        return tot
18000