<p>You wrote down many <strong>positive</strong> integers in a string called <code>num</code>. However, you realized that you forgot to add commas to seperate the different numbers. You remember that the list of integers was <strong>non-decreasing</strong> and that <strong>no</strong> integer had leading zeros.</p>

<p>Return <em>the <strong>number of possible lists of integers</strong> that you could have written down to get the string </em><code>num</code>. Since the answer may be large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

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

<pre>
<strong>Input:</strong> num = &quot;327&quot;
<strong>Output:</strong> 2
<strong>Explanation:</strong> You could have written down the numbers:
3, 27
327
</pre>

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

<pre>
<strong>Input:</strong> num = &quot;094&quot;
<strong>Output:</strong> 0
<strong>Explanation:</strong> No numbers can have leading zeros and all numbers must be positive.
</pre>

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

<pre>
<strong>Input:</strong> num = &quot;0&quot;
<strong>Output:</strong> 0
<strong>Explanation:</strong> No numbers can have leading zeros and all numbers must be positive.
</pre>

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

<ul>
	<li><code>1 &lt;= num.length &lt;= 3500</code></li>
	<li><code>num</code> consists of digits <code>&#39;0&#39;</code> through <code>&#39;9&#39;</code>.</li>
</ul>

class Solution:
    def numberOfCombinations(self, num: str) -> int:
        N = len(num)
        zeros = {i for i, v in enumerate(num) if v == '0'}

        all_one_digit = len(set(num)) == 1 and not zeros

        @cache
        def comb(i=N-1, val=float('inf'), digits=N):
            c = 0
            for j in range(max(0, i - digits + 1), i + 1):
                if j in zeros: continue
                v = int(num[j : i + 1])
                if v <= val:
                    c += 1 if j == 0 else comb(j - 1, v, i - j + 1)
            return c % int(1e9 + 7)

        if all_one_digit:
            return partitions(N) % int(1e9 + 7)

        return comb()
 
def partitions(n):
    partition_table = [1] + [0] * n

    for i in range(1, n + 1):
        for j in range(i, n + 1):
            partition_table[j] += partition_table[j - i]

    return partition_table[n]

793