<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 BIT:
    def __init__(self, n):
        self.n = n
        self.bit = [0]*(n+1)
    
    def update(self, i, x):
        i+=1 
        while i<=self.n:
            self.bit[i] += x
            i += i&-i
        
    def getSum(self, i):
        i+=1
        ans = 0
        while i>0:
            ans+= self.bit[i]
            i -= i&-i
        return ans

class Solution:
    def numTeams(self, rating: List[int]) -> int:


        bit = BIT(100011)
        n = len(rating)
        left_small_count, left_large_count = [0]*n, [0]*n
        for i in range(n):
            bit.update(rating[i], 1)
            left_small_count[i] = bit.getSum(rating[i]-1)
            left_large_count[i] = bit.getSum(100000)-bit.getSum(rating[i])
        bit.bit = [0]*100011
        for i in range(n-1, -1, -1):
            bit.update(rating[i], 1)
            left_large_count[i] *= bit.getSum(rating[i]-1)
            left_small_count[i] *= bit.getSum(100000)-bit.getSum(rating[i])
    
        return sum(left_small_count) + sum(left_large_count)

#  DP tabulization

        # n = len(rating)
        # dp_i = [[0]*4 for i in range(n+1)]
        # dp_d = [[0]*4 for i in range(n+1)]
        
        # for i in range(1, n+1):
        #     dp_i[i][1] = 1
        #     dp_d[i][1] = 1

        # for i in range( 1, n+1):
        #     for j in range(2, 4):
        #         for k in range( i):
        #             if rating[i-1]> rating[k-1]:
        #                 dp_i[i][j] += dp_i[k][j-1]
        #             if rating[i-1]< rating[k-1]:
        #                 dp_d[i][j] += dp_d[k][j-1]
        # ans = 0
        # for i in range(n+1):
        #     ans += (dp_i[i][3] + dp_d[i][3])
        # return ans

#  DP memoization

        # @cache
        # def get_increasing_count(n, size):
        #     if size > n:
        #         return 0
        #     if n==0:
        #         return 0
        #     if size==1:
        #         return 1
        #     ans = 0
            
        #     for k in range(n):
        #         if rating[n-1]>rating[k-1]:
        #             ans += get_increasing_count(k, size-1)
                
        #     return ans
        # @cache
        # def get_decreasing_count(n, size):

        #     if size > n:
        #         return 0
        #     if n==0:
        #         return 0
        #     if size==1:
        #         return 1
        #     ans = 0
        #     for k in range(n):
        #         if rating[n-1]<rating[k-1]:
        #             ans += get_decreasing_count(k, size-1)
        #     return ans
        
        # ans_inc, ans_dec =0,0
        # for i in range(n):
        #     ans_inc += get_increasing_count(i+1, 3)

        # for i in range(n):
        #     ans_dec += get_decreasing_count(i+1, 3)

        # return ans_inc + ans_dec
        

18100