<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;
    
    unordered_map<int, unordered_map<int, unsigned long>> dp;

    unsigned long func(int index, string& s, int lessNode) {
        if (s.size() == index) {
            return 1;
        }

        if (dp[index].count(lessNode)) {
            return dp[index][lessNode];
        }

        unsigned long cnt = 0;
        if (s[index] == 'D') {
            // should pick less node
            for (int i = 0; i < lessNode; i++) {
                cnt += func(index + 1, s, i);
                cnt %= MOD;
            }
        } else {
            // should pick greater node
            for (int i = 0; i < s.size() - lessNode - index; i++) {
                cnt += func(index + 1, s, lessNode + i);
                cnt %= MOD;
            }
        }

        dp[index][lessNode] = cnt;
        return cnt;
    }

    int numPermsDISequence(string s) {
        unsigned long cnt = 0;
        for (int i = 0; i <= s.size(); i++) {
            cnt += func(0, s, i);
            cnt %= MOD;
        }

        return cnt;
    }
};
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