<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 D {
        int val;

        int l_lt_cnt;
        int l_gt_cnt;

        int r_lt_cnt;
        int r_gt_cnt;
    };

    struct DComparer {
        bool operator()(const D& x, const D& y) const { return x.val < y.val; }
    };

    void F(D* beg, D* end, const DComparer& d_cmper) {
        int size{ static_cast<int>(end - beg) };

        if (size <= 1) { return; }

        int l_size{ size / 2 };
        int r_size{ size - l_size };

        D* mid{ beg + l_size };

        F(beg, mid, d_cmper);
        F(mid, end, d_cmper);

        for (int i{ 0 }, j{ 0 }; j < r_size;) {
            if (i == l_size || mid[j].val <= beg[i].val) {
                mid[j].l_lt_cnt += i;
                ++j;
            } else {
                ++i;
            }
        }

        for (int i{ l_size - 1 }, j{ r_size - 1 }; 0 <= j;) {
            if (i == -1 || beg[i].val <= mid[j].val) {
                mid[j].l_gt_cnt += l_size - 1 - i;
                --j;
            } else {
                --i;
            }
        }

        for (int i{ 0 }, j{ 0 }; i < l_size;) {
            if (j == r_size || beg[i].val <= mid[j].val) {
                beg[i].r_lt_cnt += j;
                ++i;
            } else {
                ++j;
            }
        }

        for (int i{ l_size - 1 }, j{ r_size - 1 }; 0 <= i;) {
            if (j == -1 || mid[j].val <= beg[i].val) {
                beg[i].r_gt_cnt += r_size - 1 - j;
                --i;
            } else {
                --j;
            }
        }

        std::inplace_merge(beg, mid, end, d_cmper);
    }

    int numTeams(std::vector<int>& nums) {
        int size{ static_cast<int>(nums.size()) };

        std::vector<D> ds;
        ds.resize(size);

        for (int i{ 0 }; i < size; ++i) {
            ds[i].val = nums[i];
            ds[i].l_lt_cnt = 0;
            ds[i].l_gt_cnt = 0;
            ds[i].r_lt_cnt = 0;
            ds[i].r_gt_cnt = 0;
        }

        DComparer d_cmper;

        F(&ds[0], &ds[size], d_cmper);

        int ret{ 0 };

        for (int i{ 0 }; i < size; ++i) {
            ret += ds[i].l_lt_cnt * ds[i].r_gt_cnt +
                   ds[i].l_gt_cnt * ds[i].r_lt_cnt;
        }

        return ret;
    }
};

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