Ex - Flipping Coins 2

AtCoder

IDabc272_h

Time10000ms

Memory256MB

Difficulty—

$N$ 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: $N$ $A_1$ $A_2$ $\ldots$ $A_N$ [samples]

Samples

Input #1

2
0 1

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.

Input #2

4
3 1 1 2

Output #2

665496237

Print the expected value modulo $998244353$.

API Response (JSON)

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