Score : $625$ points

问题描述

给定一个长度为 $N$ 的序列 $A = (A_1,\ldots,A_N)$。序列 $A$ 中的每个元素要么是 $-1$，要么是介于 $1$ 和 $N$ 之间的整数，并且从 $1$ 到 $N$ 的每个整数最多出现一次。

若一个排列 $P=(P_1,\ldots,P_N)$ 满足 $A_i \neq -1 \Rightarrow P_i = A_i$，则称其为 良排列。请计算所有良排列的倒序数平方和，结果对 $998244353$ 取模。

排列 $P$ 的倒序数是指满足条件 $1\leq i < j \leq N$ 且 $P_i > P_j$ 的整数对 $(i,j)$ 的数量。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a sequence $A = (A_1,\ldots,A_N)$ of length $N$. Each element of $A$ is either $-1$ or an integer between $1$ and $N$, and each integer from $1$ to $N$ is contained at most once.

A permutation $P=(P_1,\ldots,P_N)$ of $(1,\ldots,N)$ is called a good permutation if and only if it satisfies $A_i \neq -1 \Rightarrow P_i = A_i$. Find the sum of the squares of the inversion numbers of all good permutations, modulo $998244353$.

The inversion number of a permutation $P$ is the number of pairs of integers $(i,j)$ such that $1\leq i < j \leq N$ and $P_i > P_j$.

Constraints

• $1\leq N\leq 3000$
• $A_i = -1$ or $1\leq A_i \leq N$.
• Each integer from $1$ to $N$ is contained at most once in $A$.
• All input values are integers.

Input

Input is given from Standard Input in the following format:

$N$

$A_1$ $\ldots$ $A_N$

Output

Print the answer.

Sample Input 1

4
3 -1 2 -1

Sample Output 1

29

There are two good permutations: $P=(3,1,2,4)$ and $P=(3,4,2,1)$, with inversion numbers $2$ and $5$, respectively.

Thus, the answer is $2^2 + 5^2 = 29$.

Sample Input 2

10
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1

Sample Output 2

952235647

Sample Input 3

15
-1 -1 10 -1 -1 -1 2 -1 -1 3 -1 -1 -1 -1 1

Sample Output 3

460544744

Find the sum modulo $998244353$.

update @ 2024/3/10 01:14:57