<p>We call a positive integer <strong>special</strong> if all of its digits are <strong>distinct</strong>.</p>

<p>Given a <strong>positive</strong> integer <code>n</code>, return <em>the number of special integers that belong to the interval </em><code>[1, n]</code>.</p>

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

<pre>
<strong>Input:</strong> n = 20
<strong>Output:</strong> 19
<strong>Explanation:</strong> All the integers from 1 to 20, except 11, are special. Thus, there are 19 special integers.
</pre>

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

<pre>
<strong>Input:</strong> n = 5
<strong>Output:</strong> 5
<strong>Explanation:</strong> All the integers from 1 to 5 are special.
</pre>

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

<pre>
<strong>Input:</strong> n = 135
<strong>Output:</strong> 110
<strong>Explanation:</strong> There are 110 integers from 1 to 135 that are special.
Some of the integers that are not special are: 22, 114, and 131.</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 2 * 10<sup>9</sup></code></li>
</ul>

class Solution:
    def countSpecialNumbers(self, n: int) -> int:
        def count_special_numbers(num):
            digits = list(map(int, str(num)))
            n = len(digits)
            dp = [[[-1 for _ in range(1 << 10)] for _ in range(2)] for _ in range(n + 1)]
            
            def dfs(pos, tight, mask):
                if pos == n:
                    return 1 if mask != 0 else 0
                if dp[pos][tight][mask] != -1:
                    return dp[pos][tight][mask]
                
                limit = digits[pos] if tight else 9
                ans = 0
                
                for digit in range(limit + 1):
                    if (mask & (1 << digit)) == 0:
                        new_mask = mask if digit == 0 and mask == 0 else (mask | (1 << digit))
                        new_tight = tight and (digit == limit)
                        ans += dfs(pos + 1, new_tight, new_mask)
                
                dp[pos][tight][mask] = ans
                return ans
            
            return dfs(0, 1, 0)
        
        return count_special_numbers(n)
219