<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>

class Solution {
public:
    int mod = 1e9 + 7;
    int dp(string& s, int left, int right, int i, vector<vector<vector<int>>>& mem){
        if((s[i] == 'D' && left == 0) || (s[i] == 'I' && right == 0)) return 0;
        if(i == s.length()) return 1;
        if(mem[left][right][i] == -1){
            long m = 0;
            if(s[i] == 'D'){
                for(int k = 0; k < left; k++){
                    m += dp(s, k, right + left - k - 1, i + 1, mem);
                }
            }
            else{
                for(int k = 0; k < right; k++){
                    m += dp(s, left + k, right + left - (left + k) - 1, i + 1, mem);
                }
            }
            mem[left][right][i] = m % mod;
        }
        return mem[left][right][i];
    }
    int numPermsDISequence(string s) {
        int n = s.length();
        vector<vector<vector<int>>> mem(n + 2, vector<vector<int>>(n + 2, vector<int>(n + 2, -1)));
        long m = 0;
        for(int i = 0; i < n + 1; i++){
            m += dp(s, i, n - i, 0, mem);
        }
        return m % mod;
    }
};
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