Circular Dungeon

题意翻译

在游戏中，有 $n$ 个排列在环上的房间，第 $i$ 个房间只能到达第 $i+1$ 个房间（特别地，第 $n$ 个房间只能到达第 $1$ 个房间）。 同时有 $n$ 个宝箱，第 $i$ 个宝箱的价值是 $c_i$ ，其中恰好有 $k$ 个宝箱是假宝箱，其余均为真宝箱，每个房间有且仅放置一个宝箱。 玩家等概率地从 $n$ 个房间中选取一个房间开始移动，如果当前房间有真宝箱，他将获得此宝箱的价值，否则立刻结束游戏。最后获得的总收益等于之前收集的宝箱的总价值。 对于每一个 $k\in[1,n]$ ，你可以决定宝箱的排列顺序和宝箱的真假，使玩家收益的期望值最小，请求出最小的期望值 $\pmod {998244353}$。

题目描述

You are creating a level for a video game. The level consists of $ n $ rooms placed in a circle. The rooms are numbered $ 1 $ through $ n $ . Each room contains exactly one exit: completing the $ j $ -th room allows you to go the $ (j+1) $ -th room (and completing the $ n $ -th room allows you to go the $ 1 $ -st room). You are given the description of the multiset of $ n $ chests: the $ i $ -th chest has treasure value $ c_i $ . Each chest can be of one of two types: - regular chest — when a player enters a room with this chest, he grabs the treasure and proceeds to the next room; - mimic chest — when a player enters a room with this chest, the chest eats him alive, and he loses. The player starts in a random room with each room having an equal probability of being chosen. The players earnings is equal to the total value of treasure chests he'd collected before he lost. You are allowed to choose the order the chests go into the rooms. For each $ k $ from $ 1 $ to $ n $ place the chests into the rooms in such a way that: - each room contains exactly one chest; - exactly $ k $ chests are mimics; - the expected value of players earnings is minimum possible. Please note that for each $ k $ the placement is chosen independently. It can be shown that it is in the form of $ \frac{P}{Q} $ where $ P $ and $ Q $ are non-negative integers and $ Q \ne 0 $ . Report the values of $ P \cdot Q^{-1} \pmod {998244353} $ .

输入输出格式

输入格式

The first contains a single integer $ n $ ( $ 2 \le n \le 3 \cdot 10^5 $ ) — the number of rooms and the number of chests. The second line contains $ n $ integers $ c_1, c_2, \dots, c_n $ ( $ 1 \le c_i \le 10^6 $ ) — the treasure values of each chest.

输出格式

Print $ n $ integers — the $ k $ -th value should be equal to the minimum possible expected value of players earnings if the chests are placed into the rooms in some order and exactly $ k $ of the chests are mimics. It can be shown that it is in the form of $ \frac{P}{Q} $ where $ P $ and $ Q $ are non-negative integers and $ Q \ne 0 $ . Report the values of $ P \cdot Q^{-1} \pmod {998244353} $ .

输入输出样例

输入样例 #1

2
1 2

输出样例 #1

499122177 0

输入样例 #2

8
10 4 3 6 5 10 7 5

输出样例 #2

499122193 249561095 249561092 873463811 499122178 124780545 623902721 0

说明

In the first example the exact values of minimum expected values are: $ \frac 1 2 $ , $ \frac 0 2 $ . In the second example the exact values of minimum expected values are: $ \frac{132} 8 $ , $ \frac{54} 8 $ , $ \frac{30} 8 $ , $ \frac{17} 8 $ , $ \frac{12} 8 $ , $ \frac 7 8 $ , $ \frac 3 8 $ , $ \frac 0 8 $ .