CF1919E.Counting Prefixes

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题目描述

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 $998\,244\,353$ .

输入格式

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 $5000$ .

输出格式

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

输入输出样例

输入#1

5
1
0
1
1
3
-1 1 2
5
-1 0 0 1 1
5
-4 -3 -3 -2 -1

输出#1

说明/提示

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 $a = [-1, -1, -1, -1, 1]$ .

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