Ex - Flipping Coins 2
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Ex - Flipping Coins 2
Score : $600$ points
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:
Output
Print the answer.
2
0 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.
4
3 1 1 2
665496237
Print the expected value modulo $998244353$.