Inversion Squared

AtCoder

IDabc330_g

Time2000ms

Memory256MB

Difficulty—

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: $N$ $A_1$ $\ldots$ $A_N$ [samples]

Samples

Input #1

4
3 -1 2 -1

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 $2^2 + 5^2 = 29$.

Input #2

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

Output #2

952235647

Input #3

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

Output #3

460544744

Find the sum modulo $998244353$.

API Response (JSON)

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    "name": "Inversion Squared",
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      "content": "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 permu",
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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.\nA permu...",
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