<p>The <strong>beauty</strong> of a string is the difference in frequencies between the most frequent and least frequent characters.</p>

<ul>
	<li>For example, the beauty of <code>&quot;abaacc&quot;</code> is <code>3 - 1 = 2</code>.</li>
</ul>

<p>Given a string <code>s</code>, return <em>the sum of <strong>beauty</strong> of all of its substrings.</em></p>

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

<pre>
<strong>Input:</strong> s = &quot;aabcb&quot;
<strong>Output:</strong> 5
<strong>Explanation: </strong>The substrings with non-zero beauty are [&quot;aab&quot;,&quot;aabc&quot;,&quot;aabcb&quot;,&quot;abcb&quot;,&quot;bcb&quot;], each with beauty equal to 1.</pre>

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

<pre>
<strong>Input:</strong> s = &quot;aabcbaa&quot;
<strong>Output:</strong> 17
</pre>

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

<ul>
	<li><code>1 &lt;= s.length &lt;=<sup> </sup>500</code></li>
	<li><code>s</code> consists of only lowercase English letters.</li>
</ul>

class Solution:
    def beautySum(self, s: str) -> int:
        """
        beauty of a string :=
            the difference in frequencies between the most frequent and least frequent characters.

        return the sum{beauty of all of s' substrings}.
        """
        n = len(s)
        prefix_freq = [[0] * 26 for _ in range(n + 1)]

        for i, c in enumerate(s):
            prefix_freq[i + 1][ord(c) - 97] += 1
            for j in range(26):
                prefix_freq[i + 1][j] += prefix_freq[i][j]

        res = 0
        for l, r in itertools.combinations(prefix_freq, 2):
            min_freq = n
            max_freq = 0
            for j in range(26):
                freq = r[j] - l[j]
                if 0 < freq < min_freq:
                    min_freq = freq
                if freq > max_freq:
                    max_freq = freq
            res += max_freq - min_freq

        return res        
2775