Score : $600$ points

问题描述

设有编号从 $1$ 到 $N$ 的 $N$ 个球，其中第 $i$ 个球被涂成颜色 $a_i$。

对于 $(1, 2, \dots, N)$ 的一个排列 $P = (P_1, P_2, \dots, P_N)$，我们定义 $C(P)$ 如下：

• 当按顺序将球 $P_1, P_2, \dots, P_N$ 排成一行时，相邻球颜色不同的球对数量。

令 $S_N$ 表示所有 $(1, 2, \dots, N)$ 的排列集合。另外，我们定义 $F(k)$ 如下：

$\displaystyle F(k) = \left(\sum_{P \in S_N}C(P)^k \right) \mod 998244353$。

求出 $F(1), F(2), \dots, F(M)$ 的值并列出。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ balls numbered $1$ through $N$. Ball $i$ is painted in Color $a_i$.

For a permutation $P = (P_1, P_2, \dots, P_N)$ of $(1, 2, \dots, N)$, let us define $C(P)$ as follows:

• The number of pairs of adjacent balls with different colors when Balls $P_1, P_2, \dots, P_N$ are lined up in a row in this order.

Let $S_N$ be the set of all permutations of $(1, 2, \dots, N)$. Also, let us define $F(k)$ by:

$\displaystyle F(k) = \left(\sum_{P \in S_N}C(P)^k \right) \bmod 998244353$.

Enumerate $F(1), F(2), \dots, F(M)$.

Constraints

• $2 \leq N \leq 2.5 \times 10^5$
• $1 \leq M \leq 2.5 \times 10^5$
• $1 \leq a_i \leq N$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$a_1$ $a_2$ $\dots$ $a_N$

Output

Print the answers in the following format:

$F(1)$ $F(2)$ $\dots$ $F(M)$

Sample Input 1

3 4
1 1 2

Sample Output 1

8 12 20 36

Here is the list of all possible pairs of $(P, C(P))$.

• If $P=(1,2,3)$, then $C(P) = 1$.
• If $P=(1,3,2)$, then $C(P) = 2$.
• If $P=(2,1,3)$, then $C(P) = 1$.
• If $P=(2,3,1)$, then $C(P) = 2$.
• If $P=(3,1,2)$, then $C(P) = 1$.
• If $P=(3,2,1)$, then $C(P) = 1$.

We may obtain the answers by assigning these values into $F(k)$. For instance, $F(1) = 1^1 + 2^1 + 1^1 + 2^1 + 1^1 + 1^1 = 8$.

Sample Input 2

2 1
1 1

Sample Output 2

0

Sample Input 3

10 5
3 1 4 1 5 9 2 6 5 3

Sample Output 3

30481920 257886720 199419134 838462446 196874334

update @ 2024/3/10 11:02:50