Score : $600$ points

问题描述

有 $N$ 颗糖果从左至右排列成一行。
每颗糖果具有以下 $10^9$ 种颜色之一：颜色 $1$、颜色 $2$、$\ldots$、颜色 $10^9$。
对于每个 $i = 1, 2, \ldots, N$，从左边数第 $i$ 颗糖果的颜色为 Color $c_i$。

高桥将从这 $N$ 颗糖果中随机选择 $K$ 颗并获取这些糖果。
从 $N$ 颗糖果中选择 $K$ 颗的方式共有 $\binom{N}{K}$ 种，其中 $\binom{N}{K}$ 是二项式系数。高桥将以等概率随机选择这 $\binom{N}{K}$ 种方式中的一种。

由于高桥希望吃到多彩的糖果，他所选糖果的颜色种类越多，他就越开心。
针对每种情况 $K = 1, 2, \ldots, N$，计算高桥所选糖果拥有不同颜色的数量的期望值。
可以证明所求值是一个有理数。按照“注释”部分所述，将这个有理数对 $998244353$ 取模后的结果打印出来。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ candies arranged in a row from left to right.
Each candy has one of the following $10^9$ colors: Color $1$, Color $2$, $\ldots$, Color $10^9$.
For each $i = 1, 2, \ldots, N$, the $i$-th candy from the left has Color $c_i$.

Takahashi will choose $K$ from the $N$ candies and get those $K$ candies.
There are $\binom{N}{K}$ ways to choose $K$ from the $N$ candies, where $\binom{N}{K}$ is a binomial coefficient. Takahashi will randomly select one of these $\binom{N}{K}$ ways with equal probability.

Because Takahashi wants to eat colorful candies, the more variety of colors his candies have, the happier he will be.
For each scenario $K = 1, 2, \ldots, N$, find the expected value of the number of different colors Takahashi's candies will have.
We can prove that the sought value is a rational number. Print this rational number modulo $998244353$, as described in Notes.

Notes

When you print a rational number, first write it as a fraction $\frac{y}{x}$, where $x, y$ are integers and $x$ is not divisible by $998244353$ (under the constraints of the problem, such representation is always possible). Then, you need to print the only integer $z$ between $0$ and $P - 1$, inclusive, that satisfies $xz \equiv y \pmod{998244353}$.

Constraints

• $1 \leq N \leq 5 \times 10^4$
• $1 \leq c_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$c_1$ $c_2$ $\ldots$ $c_N$

Output

Print $N$ lines. The $i$-th line should contain the expected value of the number of different colors Takahashi's candies will have for the scenario $K = i$, modulo $998244353$ as described in Notes.

Sample Input 1

3
1 2 2

Sample Output 1

1
665496237
2

When $K = 1$, he will get the $1$-st candy, $2$-nd candy, or $3$-rd candy. In any case, his candy will have one color, so the expected value of the number of different colors is $1$.

When $K = 2$, he will get the $1$-st and $2$-nd candies, $2$-nd and $3$-rd candies, or $1$-st and $3$-rd candies.

• If he gets the $1$-st and $2$-nd candies, they have two different colors.
• If he gets the $2$-nd and $3$-rd candies, they have one color.
• If he gets the $1$-st and $3$-rd candies, they have two different colors.

Thus, the expected value of the number of different colors is $\frac{1}{3} \cdot 2 + \frac{1}{3} \cdot 1 + \frac{1}{3} \cdot 2 = \frac{5}{3}$.
Be sure to print it modulo $998244353$ as described in Notes.

When $K = 3$, he will always get the $1$-st, $2$-nd, and $3$-rd candies, which have two different colors, so the expected value of the number of different colors is $2$.

Sample Input 2

11
3 1 4 1 5 9 2 6 5 3 5

Sample Output 2

1
725995895
532396991
768345657
786495555
937744700
574746754
48399732
707846002
907494873
7

update @ 2024/3/10 09:33:43