<p>You are given a string <code>s</code> of length <code>n</code> where <code>s[i]</code> is either:</p>

<ul>
	<li><code>&#39;D&#39;</code> means decreasing, or</li>
	<li><code>&#39;I&#39;</code> means increasing.</li>
</ul>

<p>A permutation <code>perm</code> of <code>n + 1</code> integers of all the integers in the range <code>[0, n]</code> is called a <strong>valid permutation</strong> if for all valid <code>i</code>:</p>

<ul>
	<li>If <code>s[i] == &#39;D&#39;</code>, then <code>perm[i] &gt; perm[i + 1]</code>, and</li>
	<li>If <code>s[i] == &#39;I&#39;</code>, then <code>perm[i] &lt; perm[i + 1]</code>.</li>
</ul>

<p>Return <em>the number of <strong>valid permutations</strong> </em><code>perm</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> s = &quot;DID&quot;
<strong>Output:</strong> 5
<strong>Explanation:</strong> The 5 valid permutations of (0, 1, 2, 3) are:
(1, 0, 3, 2)
(2, 0, 3, 1)
(2, 1, 3, 0)
(3, 0, 2, 1)
(3, 1, 2, 0)
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;D&quot;
<strong>Output:</strong> 1
</pre>

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

<ul>
	<li><code>n == s.length</code></li>
	<li><code>1 &lt;= n &lt;= 200</code></li>
	<li><code>s[i]</code> is either <code>&#39;I&#39;</code> or <code>&#39;D&#39;</code>.</li>
</ul>

from collections import defaultdict

class Solution:
    def numPermsDISequence(self, s: str) -> int:
        # Memoization dictionary to store results of subproblems
        mem = defaultdict(int)
        
        # Helper function to perform depth-first search
        def dfs(i, val=0):
            # Base case: if we have processed the entire string, return 1
            if i == len(s):
                return 1
            
            # Check if the result for this subproblem is already computed
            if (i, val) in mem:
                return mem[i, val]
            
            # Initialize the number of valid permutations for this subproblem
            p = 0
            
            # If the current character is 'D', we need to ensure decreasing order
            if s[i] == "D":
                # Iterate over possible values for the next position
                for ind in range(0, val + 1):
                    # Recursively calculate permutations for the next position
                    p += dfs(i + 1, ind) % (10**9 + 7)
            
            # If the current character is 'I', we need to ensure increasing order
            else:
                # Iterate over possible values for the next position
                for ind in range(val + 1, i + 2):
                    # Recursively calculate permutations for the next position
                    p += dfs(i + 1, ind) % (10**9 + 7)
            
            # Store the result in the memoization dictionary
            mem[i, val] = p
            return p
        
        # Start the DFS from the first character with initial value 0
        return dfs(0)

# Explanation of the solution approach:
# 1. We use a recursive depth-first search (DFS) approach with memoization to explore all possible permutations.
# 2. The function `dfs(i, val)` calculates the number of valid permutations starting from index `i` with the last used value `val`.
# 3. If the current character in the string `s` is 'D', we ensure that the next number in the permutation is less than or equal to `val`.
# 4. If the current character is 'I', we ensure that the next number is greater than `val`.
# 5. We use a memoization dictionary `mem` to store results of subproblems to avoid redundant calculations.
# 6. The result is computed modulo 10^9 + 7 to handle large numbers.
26200