<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 {
public:
    struct Item {
        int value;
        int index;
    };
    
    int numTeams(vector<int>& rating) {
        auto n = rating.size();
        
        vector<int> lns(n, 0);
        vector<int> rns(n, 0);
        
        vector<Item> values(n);
        
        for (size_t i = 0; i < n; i++) {
            values[i].value = rating[i];
            values[i].index = i;
        }
        
        sort(values, lns, rns, 0, n);
        
        long long count = 0;
        for (int i = 0; i < n; i++) {
            count += lns[i] * rns[i];
            count += (i - lns[i]) * (n - i - 1 - rns[i]);
        }
        
        return count;
    }
    void merge(vector<Item> & values, vector<int> &lns, vector<int> &rns, int left, int middle, int right) {
        int ln = middle - left;
        int rn = right - middle;

        vector<Item> ls(ln);
        vector<Item> rs(rn);

        for(int i = left; i < middle; i++){
            ls[i - left] = values[i];
        }
        for(int i = middle; i < right; i++){
            rs[i - middle] = values[i];
        }

        int i = left;
        int l = 0;
        int r = 0;

        while(l < ln && r < rn){
            if (ls[l].value < rs[r].value){
                values[i] = ls[l++];
                rns[values[i].index] += r;
            }
            else {
                values[i] = rs[r++];
                lns[values[i].index] += middle - left - l;
            }

            i++;
        }

        while(l < ln){
            values[i] = ls[l++];
            rns[values[i++].index] += rn;
        }
        while(r < rn){
            values[i] = rs[r++];
            lns[values[i++].index] += middle - left - ln;
        }
    }

    void sort(vector<Item> &values, vector<int> &lns, vector<int> &rb, int left, int right) {
        if (right - left <= 1) {
            return;
        }

        auto middle = left + (right - left) / 2;

        sort(values, lns, rb, left, middle);
        sort(values, lns, rb, middle, right);
        merge(values, lns, rb, left, middle, right);
    }
};
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