<p>You are given a <strong>0-indexed</strong> integer array <code>nums</code>. For each index <code>i</code> (<code>1 &lt;= i &lt;= nums.length - 2</code>) the <strong>beauty</strong> of <code>nums[i]</code> equals:</p>

<ul>
	<li><code>2</code>, if <code>nums[j] &lt; nums[i] &lt; nums[k]</code>, for <strong>all</strong> <code>0 &lt;= j &lt; i</code> and for <strong>all</strong> <code>i &lt; k &lt;= nums.length - 1</code>.</li>
	<li><code>1</code>, if <code>nums[i - 1] &lt; nums[i] &lt; nums[i + 1]</code>, and the previous condition is not satisfied.</li>
	<li><code>0</code>, if none of the previous conditions holds.</li>
</ul>

<p>Return<em> the <strong>sum of beauty</strong> of all </em><code>nums[i]</code><em> where </em><code>1 &lt;= i &lt;= nums.length - 2</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [1,2,3]
<strong>Output:</strong> 2
<strong>Explanation:</strong> For each index i in the range 1 &lt;= i &lt;= 1:
- The beauty of nums[1] equals 2.
</pre>

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

<pre>
<strong>Input:</strong> nums = [2,4,6,4]
<strong>Output:</strong> 1
<strong>Explanation:</strong> For each index i in the range 1 &lt;= i &lt;= 2:
- The beauty of nums[1] equals 1.
- The beauty of nums[2] equals 0.
</pre>

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

<pre>
<strong>Input:</strong> nums = [3,2,1]
<strong>Output:</strong> 0
<strong>Explanation:</strong> For each index i in the range 1 &lt;= i &lt;= 1:
- The beauty of nums[1] equals 0.
</pre>

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

<ul>
	<li><code>3 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>5</sup></code></li>
</ul>

from typing import List

class Solution:
    def sumOfBeauties(self, nums: List[int]) -> int:
        n = len(nums)
        left_max = [nums[0]] * n
        right_min = [nums[-1]] * n
        beauty_sum = 0

        # Precompute left maximums
        for i in range(1, n):
            left_max[i] = max(left_max[i - 1], nums[i])

        # Precompute right minimums
        for i in range(n - 2, -1, -1):
            right_min[i] = min(right_min[i + 1], nums[i])

        # Calculate beauty for each element
        for i in range(1, n - 1):
            if left_max[i - 1] < nums[i] < right_min[i + 1]:
                beauty_sum += 2
            elif nums[i - 1] < nums[i] < nums[i + 1]:
                beauty_sum += 1

        return beauty_sum

# Example usage:
solution = Solution()
nums = [1, 2, 3]
print(solution.sumOfBeauties(nums))  # Output: 2

nums = [2, 4, 6, 4]
print(solution.sumOfBeauties(nums))  # Output: 1

nums = [3, 2, 1]
print(solution.sumOfBeauties(nums))  # Output: 0
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