A16388 | $Counting Prefixes$

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内存限制256MB

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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 $998\,244\,353$ .

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 $5000$ .

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

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

C++ 编辑器

输入

输出