Score : $600$ points

问题描述

设有 $N$ 个编号从 $1$ 到 $N$ 的球。最初，第 $i$ 个球被涂上了颜色 $A_i$。

颜色用 $1$ 到 $N$ 之间的整数表示（包括 $1$ 和 $N$）。

考虑重复执行以下操作，直到所有球的颜色变得相同。

• 在包含 $N$ 个球的集合的所有 $2^N$ 个子集（包括空集）中，均匀随机选择一个子集。设所选球的索引为 $X_1, X_2, ..., X_K$。接下来，通过从整数 $(1,2,\dots,N)$ 中均匀随机选择 $K$ 个整数得到一个排列。设所选排列为 $P=(P_1,P_2,\dots,P_K)$。对于满足 $1 \le i \le K$ 的每个整数 $i$，将第 $X_i$ 个球的颜色改为 $P_i$。

求出执行操作次数的期望值，结果对 $998244353$ 取模。

这里所说的通过从整数 $(1,2,\dots,N)$ 中选择 $K$ 个整数得到的排列是指一个包含 $K$ 个在 $1$ 到 $N$ 之间（包括 $1$ 和 $N$）的互不相同的整数构成的序列。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ balls numbered $1$ through $N$. Initially, Ball $i$ is painted in Color $A_i$.

Colors are represented by integers between $1$ and $N$, inclusive.

Consider repeating the following operation until all the colors of the balls become the same.

• There are $2^N$ subsets (including the empty set) of the set consisting of the $N$ balls; choose one of the subsets uniformly at random. Let $X_1,X_2,...,X_K$ be the indices of the chosen balls. Next, choose a permutation obtained by choosing $K$ integers out of integers $(1,2,\dots,N)$ uniformly at random. Let $P=(P_1,P_2,\dots,P_K)$ be the chosen permutation. For each integer $i$ such that $1 \le i \le K$, change the color of Ball $X_i$ to $P_i$.

Find the expected value of number of operations, modulo $998244353$.

Here a permutation obtained by choosing $K$ integers out of integers $(1,2,\dots,N)$ is a sequence of $K$ integers between $1$ and $N$, inclusive, whose elements are pairwise distinct.

Notes

We can prove that the sought expected value is always a rational number. Also, under the Constraints of this problem, when the value is represented by two coprime integers $P$ and $Q$ as $\frac{P}{Q}$, we can prove that there exists a unique integer $R$ such that $R \times Q \equiv P(\bmod\ 998244353)$ and $0 \le R < 998244353$. Find this $R$.

Constraints

• $2 \le N \le 2000$
• $1 \le A_i \le N$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\dots$ $A_N$

Output

Print the answer.

Sample Input 1

2
1 2

Sample Output 1

4

The operation is repeated until a subset of size $1$ is chosen and the color of the ball is changed to the color of the ball not contained in the subset. The probability is $\displaystyle \frac{2}{4} \times \frac{1}{2}=\frac{1}{4}$, so the expected value is $4$.

Sample Input 2

3
1 1 1

Sample Output 2

0

The operation is never performed, since the colors of the balls are already the same.

Sample Input 3

10
3 1 4 1 5 9 2 6 5 3

Sample Output 3

900221128

update @ 2024/3/10 10:40:16