<p>Given an integer array <code>nums</code>, return<em> an integer array </em><code>counts</code><em> where </em><code>counts[i]</code><em> is the number of smaller elements to the right of </em><code>nums[i]</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [5,2,6,1]
<strong>Output:</strong> [2,1,1,0]
<strong>Explanation:</strong>
To the right of 5 there are <b>2</b> smaller elements (2 and 1).
To the right of 2 there is only <b>1</b> smaller element (1).
To the right of 6 there is <b>1</b> smaller element (1).
To the right of 1 there is <b>0</b> smaller element.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [-1]
<strong>Output:</strong> [0]
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [-1,-1]
<strong>Output:</strong> [0,0]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>-10<sup>4</sup> &lt;= nums[i] &lt;= 10<sup>4</sup></code></li>
</ul>

def lsb(x: int) -> int:
    return x & -x

class BIT:
    def __init__(self, length: int) -> None:
        self.array = [0 for _ in range(length + 1)]
    
    def query(self, key: int) -> int:
        """
        Returns the sum of elements in the range [0, key]
        """
        key += 1
        result = 0
        while key > 0:
            result += self.array[key]
            key -= lsb(key)
        return result
    
    def update(self, key: int, value: int) -> None:
        """
        Increases the number at key by vlaue
        """
        key += 1
        while key < len(self.array):
            self.array[key] += value
            key += lsb(key)

class Solution:
    def countSmaller(self, nums: List[int]) -> List[int]:
        l = len(nums)
        sorted_nums = sorted((num, i) for i, num in enumerate(nums))

        # Each i stores the count of larger numbers in [i + 1, len(nums)]
        bit = BIT(l)

        result = [0 for _ in nums]

        for num, i in sorted_nums:
            key = (l - 1) - i

            result[i] = bit.query(key)
            bit.update(key, 1)
        
        return result

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