CF1919E.Counting Prefixes

There is a hidden array $a$ of size $n$ consisting of only $1$ and $-1$. Let $p$ be the prefix sums of array $a$. More formally, $p$ is an array of length $n$ defined as $p_i = a_1 + a_2 + \ldots + a_i$. Afterwards, array $p$ is sorted in non-decreasing order. For example, if $a = [1, -1, -1, 1, 1]$, then $p = [1, 0, -1, 0, 1]$ before sorting and $p = [-1, 0, 0, 1, 1]$ after sorting.

You are given the prefix sum array $p$ after sorting, but you do not know what array $a$ is. Your task is to count the number of initial arrays $a$ such that the above process results in the given sorted prefix sum array $p$. As this number can be large, you are only required to find it modulo $998\,244\,353$.

存在一个长度为 $n$ 的隐藏数组 $a$，其中仅包含 $1$ 和 $-1$。令 $p$ 表示数组 $a$ 的前缀和数组。更准确地说，$p$ 是一个长度为 $n$ 的数组，定义为 $p_i = a_1 + a_2 + \ldots + a_i$。随后，数组 $p$ 按非递减顺序排序。例如，若 $a = [1, -1, -1, 1, 1]$，则排序前 $p = [1, 0, -1, 0, 1]$，排序后 $p = [-1, 0, 0, 1, 1]$。

你被给定排序后的前缀和数组 $p$，但并不知道原始数组 $a$ 是什么。你的任务是计算满足上述过程能产生给定的已排序前缀和数组 $p$ 的初始数组 $a$ 的个数。由于该数目可能很大，你只需输出其对 $998\,244\,353$ 取模的结果。

Each test contains multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 1000$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1 \le n \le 5000$) — the size of the hidden array $a$.

The second line of each test case contains $n$ integers $p_1, p_2, \ldots, p_n$ ($|p_i| \le n$) — the $n$ prefix sums of $a$ sorted in non-decreasing order.

It is guaranteed that $p_1 \le p_2 \le \ldots \le p_n$.

It is guaranteed that the sum of $n$ over all test cases does not exceed $5000$.

每个测试包含多个测试用例。第一行包含一个整数 $t$（$1 \leq t \leq 1000$），表示测试用例的数量。随后是各测试用例的描述。

每个测试用例的第一行包含一个整数 $n$（$1 \le n \le 5000$），表示隐藏数组 $a$ 的大小。

每个测试用例的第二行包含 $n$ 个整数 $p_1, p_2, \ldots, p_n$（$|p_i| \le n$），即数组 $a$ 的 $n$ 个前缀和按非递减顺序排序后的结果。

保证有 $p_1 \le p_2 \le \ldots \le p_n$。

保证所有测试用例的 $n$ 值之和不超过 $5000$。

For each test case, output the answer modulo $998\,244\,353$.

对于每个测试用例，输出答案对 $998\,244\,353$ 取模的结果。

In the first two test cases, the only possible arrays $a$ for $n = 1$ are $a = [1]$ and $a = [-1]$. Their respective sorted prefix sum arrays $p$ are $p = [1]$ and $p = [-1]$. Hence, there is no array $a$ that can result in the sorted prefix sum array $p = [0]$ and there is exactly $1$ array $a$ that can result in the sorted prefix sum array $p = [1]$.

In the third test case, it can be proven that there is no array $a$ that could result in the sorted prefix sum array $p = [-1, 1, 2]$.

In the fourth test case, the $3$ possible arrays $a$ that could result in the sorted prefix sum array $p = [-1, 0, 0, 1, 1]$ are:

• $a = [1, -1, 1, -1, -1]$. The prefix sum array before sorting is $p = [1, 0, 1, 0, -1]$, which after sorting gives $p = [-1, 0, 0, 1, 1]$.
• $a = [1, -1, -1, 1, 1]$. The prefix sum array before sorting is $p = [1, 0, -1, 0, 1]$, which after sorting gives $p = [-1, 0, 0, 1, 1]$.
• $a = [-1, 1, 1, -1, 1]$. The prefix sum array before sorting is $p = [-1, 0, 1, 0, 1]$, which after sorting gives $p = [-1, 0, 0, 1, 1]$.

For the fifth test case, the only possible array $a$ that could result in the sorted prefix sum array $p = [-4, -3, -3, -2, -1]$ is $a = [-1, -1, -1, -1, 1]$.

在前两个测试用例中，当 $n = 1$ 时，唯一可能的数组 $a$ 分别为 $a = [1]$ 和 $a = [-1]$。它们各自对应的排序后前缀和数组 $p$ 分别为 $p = [1]$ 和 $p = [-1]$。因此，不存在任何数组 $a$ 能够得到排序后的前缀和数组 $p = [0]$；而恰好存在 $1$ 个数组 $a$ 能够得到排序后的前缀和数组 $p = [1]$。

在第三个测试用例中，可以证明：不存在任何数组 $a$ 能够得到排序后的前缀和数组 $p = [-1, 1, 2]$。

在第四个测试用例中，能够得到排序后的前缀和数组 $p = [-1, 0, 0, 1, 1]$ 的 $3$ 个可能的数组 $a$ 为：

• $a = [1, -1, 1, -1, -1]$。排序前的前缀和数组为 $p = [1, 0, 1, 0, -1]$，排序后得到 $p = [-1, 0, 0, 1, 1]$。
• $a = [1, -1, -1, 1, 1]$。排序前的前缀和数组为 $p = [1, 0, -1, 0, 1]$，排序后得到 $p = [-1, 0, 0, 1, 1]$。
• $a = [-1, 1, 1, -1, 1]$。排序前的前缀和数组为 $p = [-1, 0, 1, 0, 1]$，排序后得到 $p = [-1, 0, 0, 1, 1]$。

在第五个测试用例中，唯一可能的数组 $a$，使得其排序后的前缀和数组为 $p = [-4, -3, -3, -2, -1]$，是 $a = [-1, -1, -1, -1, 1]$。