We are given S, a length n string of characters from the set {'D', 'I'}. (These letters stand for "decreasing" and "increasing".)
A valid permutation is a permutation P[0], P[1], ..., P[n] of integers {0, 1, ..., n}, such that for all i:

If S[i] == 'D', then P[i] > P[i+1], and;
If S[i] == 'I', then P[i] < P[i+1].

How many valid permutations are there?  Since the answer may be large, return your answer modulo 10^9 + 7.
 
Example 1:
Input: "DID"
Output: 5
Explanation: 
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)

 
Note:

1 <= S.length <= 200
S consists only of characters from the set {'D', 'I'}.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = "from functools import lru_cache\nclass Solution:\n    def numPermsDISequence(self, S: str) -> int:\n        '''\n        这道题目的关键是第一步看到10**9+7知道用dp方法来做\n        dp[i]表示范围[0，i]的序列个数，然后发现不好找递推关系\n        这个时候肯定是隐藏的信息没用，外面需要通过实际的例子来总结影响个数的另外一个参数\n        序列是升还是降跟什么关系最大，那就是最后一个数字。DID模式，我们可以是1032，那么我们再加个D，要求递减\n        这个时候多出了一个4来，我们如何改造1032呢，直接加4是不行的，那么加一样的2？改成10432，相当于3，2都+1\n        加1呢？改成20431，相当于1，3，2都+1，那么规律是什么。\n        如果是D，那么只能放<=最后一个数的数，然后把原来所有>=尾巴的数都+1,最后一个数很重要\n        反过来想，如果是I,那么只能放>最后一个数的数，然后原来所有的>=新加的数都+1\n        '''\n        mod = 10 ** 9 + 7\n        n = len(S)\n        \n        @lru_cache(None)\n        def dfs(i, j):  # 位置i处数字j结尾 数字j的范围是0-i\n            if i == 0:\n                return 1\n            elif S[i - 1] == 'D':\n                return sum(dfs(i - 1, k) for k in range(j, i)) % mod\n            else:\n                return sum(dfs(i - 1, k) for k in range(j)) % mod\n        return sum(dfs(n, j) for j in range(n + 1)) % mod"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'numPermsDISequence'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 13898