<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 Solution {
private:
    vector<int> mBIT;
    int n = 0;
    
    int query(int index) {  // 1-indexed
        int sum = 0;
        for (int i = index; i > 0; i -= i & -i) {
            sum += mBIT[i];
        }               // 0 1 2 3 4 5 6
        return sum;     //     
    }
    
    void update(int index, int delta) {
        for (int i = index; i < n + 1; i += i & -i) {
            mBIT[i] += delta;
        }
    }
    
    int rangeSum(int left, int right) {
        return query(right) - query(left - 1);
    }
    
public:
    int numTeams(vector<int>& rating) {
        // [2,5,3,4,1]
        //  0 1 1 2 0
        //  3 0 1 0 0
        
        //  0 0 1 1 4
        //  1 3 1 1 0
        int res = 0;
        int maxEle = *ranges::max_element(rating);
        n = *ranges::max_element(rating) + 3;  // 多加右邊的 sentinel
        mBIT.resize(n + 1);                    //     左
        vector<int> leftLess(rating.size());
        
        for (int i = 0; i < rating.size(); ++i) {
            leftLess[i] = query(rating[i] - 1);
            update(rating[i], 1);
        }
        
        for (int i = 0; i < mBIT.size(); ++i) {
            mBIT[i] = 0;
        }
        
        for (int i = rating.size() - 1; i >= 0; --i) {
            int rightGreater = rangeSum(rating[i] + 1, maxEle + 1);
            int increasing = leftLess[i] * rightGreater;
            
            int leftGreater = i - leftLess[i];
            int rightLess = rating.size() - 1 - i - rightGreater;
            int decreasing = leftGreater * rightLess;
            
            res += (increasing + decreasing);
            
            update(rating[i], 1);
        }
        
        return res;
    }
};
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