<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:
    vector<int> LS, RG, LG, RS;
    void merge(int l, int m, int r, vector<pair<int, int>> &nums) {
        vector<pair<int, int>> L(nums.begin() + l, nums.begin() + m + 1);
        vector<pair<int, int>> R(nums.begin() + m + 1, nums.begin() + r + 1);
        int i = 0, j = 0, k = l, cnt = 0, cnt1 = 0;     
        while(i < L.size() && j < R.size()) {
            if(L[i].first < R[j].first) {
                cnt++;
                RS[L[i].second] += cnt1;
                RG[L[i].second] += R.size() - j;
                nums[k++] = L[i++];
            }
            else {
                cnt1++;
                LG[R[j].second] += L.size() - i;
                LS[R[j].second] += cnt;
                nums[k++] = R[j++];
            }
        }
        while(i < L.size()) {
            RS[L[i].second] += cnt1;
            nums[k++] = L[i++];           
        }
        while(j < R.size()) {
            LS[R[j].second] += cnt;
            nums[k++] = R[j++];
        }
    }
    void mergeSort(int l, int r, vector<pair<int, int>> &nums) {
        if(l >= r) return;
        int m = (l + r) / 2;
        mergeSort(l, m, nums);
        mergeSort(m + 1, r, nums);
        
        merge(l, m, r, nums);
    }
    int numTeams(vector<int>& a) {
        int n = a.size();
        LS.resize(n);
        RS.resize(n);
        LG.resize(n);
        RG.resize(n);
        LS.assign(n, 0);
        RS.assign(n, 0);
        LG.assign(n, 0);
        RG.assign(n, 0);
        vector<pair<int, int>> v(n);
        for(int i = 0; i < n; i++) v[i] = {a[i], i};
        mergeSort(0, n - 1, v);
        int ans = 0;
        // for(int i = 0; i < n; i++) {
        //     cout<<LS[i]<<" "<<LG[i]<<" "<<RS[i]<<" "<<RG[i]<<"\n";
        // }
        for(int i = 1; i < n - 1; i++) {
            ans += LS[i] * RG[i] + LG[i] * RS[i];
        }
        return ans;
    }
};


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