<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>

/*
given x find #{ratings[i] > x}
*/
struct Fenwick {
    Fenwick(int n): n(n) {buffer = vector<int>(n, 0);}

    // get number of observed buffer which <= x
    int query(int x) const {
        int tot = 0;
        while (x >= 0) {
            tot += buffer[x];

            x = (x & (x+1)) - 1;
        }
        return tot;
    }

    // insert x to observed value
    void insert(int x) {
        while (x < n) {
            buffer[x] += 1;
            x = x | (x+1);
        }
    }

    vector<int> buffer;
    int n; // n == max_value + 1
};

class Solution {
public:
    int numTeams(vector<int>& ratings) {
        int max_rating = 0;
        for (int rating : ratings)
            max_rating = max(max_rating, rating);
        
        Fenwick tree(max_rating+1);
        tree.insert(ratings[0]);

        int n = ratings.size();
        int result = 0;
        for (int i=1; i < n; ++i) {
            int num_greater = 0; // (r[i] < r[j])
            int num_less = 0; // (r[i] > r[j])
            for (int j=i+1; j < n; ++j) {
                num_less += ratings[i] > ratings[j];
                num_greater += ratings[i] < ratings[j];
            }

            // find r[h] < r[i]
            int num_greater_prev = tree.query(ratings[i]);
            int num_less_prev = (i-num_greater_prev);
            
            // printf("i=%d\n", i);
            // printf("r[h] < r[i] = %d, r[i] < r[j] = %d\n", num_greater_prev, num_greater);
            // printf("r[h] > r[i] = %d, r[i] > r[j] = %d\n", num_less_prev, num_less);

            result += num_greater * num_greater_prev + num_less_prev * num_less;
            tree.insert(ratings[i]);
        }

        return result;
    }
};
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