Given an array A of non-negative integers, the array is squareful if for every pair of adjacent elements, their sum is a perfect square.
Return the number of permutations of A that are squareful.  Two permutations A1 and A2 differ if and only if there is some index i such that A1[i] != A2[i].
 
Example 1:
Input: [1,17,8]
Output: 2
Explanation: 
[1,8,17] and [17,8,1] are the valid permutations.

Example 2:
Input: [2,2,2]
Output: 1

 
Note:

1 <= A.length <= 12
0 <= A[i] <= 1e9

```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 collections import Counter\nfrom functools import lru_cache\nimport math\n\nclass Solution:\n    def numSquarefulPerms(self, A: List[int]) -> int:\n        # Backtracking\n        # Time  complexity: O(N^N)\n        # Space complexity: O(N)\n        # N = len(A)\n        # count = Counter(A)\n\n        # graph = {x: [] for x in count}\n        # for x in count:\n        #     for y in count:\n        #         if int((x + y) ** .5 + 0.5) ** 2 == x + y:\n        #             graph[x].append(y)\n\n        # def dfs(x, todo):\n        #     count[x] -= 1\n        #     if todo == 0:\n        #         ans = 1\n        #     else:\n        #         ans = 0\n        #         for y in graph[x]:\n        #             if count[y]:\n        #                 ans += dfs(y, todo - 1)\n\n        #     count[x] += 1\n        #     return ans\n\n        # return sum(dfs(x, len(A) - 1) for x in count)\n\n\n        # Dynamic Programming\n        # Time  complexity: O(N^2 x 2^N) = O(sum_{k=1}^{N} C_N^K x C_k^2)\n        # Space complexity: O(N x 2^N)\n        N = len(A)\n\n        def edge(x, y):\n            r = math.sqrt(x + y)\n            return int(r + 0.5) ** 2 == x + y\n\n        graph = [[] for _ in range(len(A))]\n        for i, x in enumerate(A):\n            for j in range(i):\n                if edge(x, A[j]):\n                    graph[i].append(j)\n                    graph[j].append(i)\n\n        # find num of hamiltonian paths in graph\n        @lru_cache(None)\n        def dfs(node, visited):\n            if visited == (1 << N) - 1:\n                return 1\n\n            ans = 0\n            for nei in graph[node]:\n                if (visited >> nei) & 1 == 0:\n                    ans += dfs(nei, visited | 1 << nei)\n            return ans\n\n        ans = sum(dfs(i, 1 << i) for i in range(N))\n        count = Counter(A)\n        for v in list(count.values()):\n            ans //= math.factorial(v)\n        return ans\n\n\n        # c = Counter(A)\n        # cand = {i: {j for j in c if int((i+j)**.5) ** 2 == i + j} for i in c}\n        # self.res = 0\n\n        # def dfs(x, left=len(A)-1):\n        #     c[x] -= 1\n        #     if left == 0:\n        #         self.res += 1\n        #     for y in cand[x]:\n        #         if c[y]:\n        #             dfs(y, left - 1)\n        #     c[x] += 1\n\n        # for x in c: dfs(x)\n        # return self.res\n\n\n        # def backtrack(i, A):\n        #     if i > 1:\n        #         val = A[i - 2] + A[i - 1]\n        #         is_square = math.sqrt(val) == math.floor(math.sqrt(val))\n        #         if not is_square:\n        #             return\n\n        #     if i == len(A):\n        #         output.append(A)\n        #         return\n\n        #     for j in range(i, len(A)):\n        #         A[i], A[j] = A[j], A[i]\n        #         search = (i, A[i], tuple(A[:i]))\n        #         if search not in seen:\n        #             seen.add(search)\n        #             backtrack(i + 1, A)\n\n        #         A[i], A[j] = A[j], A[i]\n\n        # output, seen = [], set()\n        # backtrack(0, A)\n        # return len(output)\n\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'numSquarefulPerms'
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: 13822