Score : $600$ points

问题描述

设有编号为 $0,1,\ldots,N-1$ 的 $N$ 枚硬币排成一行。最初，所有硬币均为正面朝上。同时，你得到了一个长度为 $N$ 的由 $0$ 到 $N-1$ 的整数组成的序列 $A$。

Snuke 将以等概率选择一个包含 $(1,\ldots,N)$ 的排列 $p=(p_1,p_2,\ldots,p_N)$，并执行 $N$ 次操作。在第 $i$ 次 $(1\leq i \leq N)$ 操作中，

• 他会翻转 $(A_{p_i}+1)$ 枚硬币：编号为 $(i-1) \bmod N$、$(i-1+1 ) \bmod N$、$\ldots$ 和 $(i -1+ A_{p_i}) \bmod N$ 的硬币。

经过 $N$ 次操作后，Snuke 从他母亲那里得到 $k$ 日元（日本货币），其中 $k$ 是正面朝上的硬币数量。

求 Snuke 将会收到的钱数的期望值，模 $998244353$。

模 $998244353$ 下期望值的定义

在这个问题中，可以证明所求期望值始终是一个有理数。此外，在本题的约束条件下，若所求期望值表示为不可约分数 $\frac{y}{x}$，则保证 $x$ 不被 $998244353$ 整除。

此时，存在一个介于 $0$ 和 $998244352$ 之间的整数 $z$，满足 $xz \equiv y \pmod{998244353}$，且这个 $z$ 唯一确定。找出这样的 $z$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

$N$ coins numbered $0,1,\ldots,N-1$ are arranged in a row. Initially, all coins are face up. Also, you are given a sequence $A$ of length $N$ consisting of integers between $0$ and $N-1$.

Snuke will choose a permutation $p=(p_1,p_2,\ldots,p_N)$ of $(1,\ldots,N)$ at equal probability and perform $N$ operations. In the $i$-th $(1\leq i \leq N)$ operation,

• he flips $(A_{p_i}+1)$ coins: coin $(i-1) \bmod N$, coin $(i-1+1 ) \bmod N$, $\ldots$, and coin $(i -1+ A_{p_i}) \bmod N$.

After the $N$ operations, Snuke receives $k$ yen (the currency in Japan) from his mother, where $k$ is the number of face-up coins.

Find the expected value, modulo $998244353$, of the money Snuke will receive.

Definition of expected value modulo $998244353$

In this problem, we can prove that the sought expected value is always a rational number. Moreover, under the Constraints of this problem, when the sought expected value is represented as an irreducible fraction $\frac{y}{x}$, it is guaranteed that $x$ is indivisible by $998244353$.

Then, an integer $z$ between $0$ and $998244352$ such that $xz \equiv y \pmod{998244353}$ is uniquely determined. Find such $z$.

Constraints

• $1 \leq N\leq 2\times 10^5$
• $0\leq A_i \leq N-1$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print the answer.

Sample Input 1

2
0 1

Sample Output 1

1

$p$ can be either $(1,2)$ or $(2,1)$.

• If $(1,2)$ is chosen as $p$:

In the first operation, coin $0$ is flipped, and in the second operation, coin $1$ and coin $0$ are flipped. One coin, coin $0$, results in being face up, so he receives $1$ yen.

• If $(2,1)$ is chosen as $p$:

In the first operation, coin $0$ and coin $1$ are flipped, and in the second operation, coin $1$ is flipped. One coin, coin $1$, results in being face up, so he receives $1$ yen.

Therefore, the expected value of the money he receives is $1$ yen.

Sample Input 2

4
3 1 1 2

Sample Output 2

665496237

Print the expected value modulo $998244353$.

update @ 2024/3/10 11:29:18