Score : $600$ points

问题描述

有$N$个人，依次编号为Person $1$、Person $2$、$\dots$、Person $N$，他们按照从前往后的顺序$(1,2,\dots,N)$排成一排。其中，第$i$个人穿着颜色$c_i$。

高桥重复了以下操作$K$次：任意选择两个人$i$和$j$，交换Person $i$和Person $j$的位置。

在完成$K$次操作后，对于所有满足$1 \leq i \leq N$的整数$i$，从前往数第$i$个人所穿的颜色与$c_i$相同。

请问，在$K$次操作结束后，有多少种可能的人排列方式？请计算这个排列数对$998244353$取模的结果。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ people called Person $1$, Person $2$, $\dots$, Person $N$, lined up in a row in the order of $(1,2,\dots,N)$ from the front. Person $i$ is wearing Color $c_i$.
Takahashi repeated the following operation $K$ times: choose two People $i$ and $j$ arbitrarily and swap the positions of Person $i$ and Person $j$.
After the $K$ operations have ended, the color that the $i$-th person from the front is wearing coincided with $c_i$, for every integer $i$ such that $1 \leq i \leq N$.
How many possible permutations of people after the $K$ operations are there? Find the count modulo $998244353$.

Constraints

• $2 \leq N \leq 200000$
• $1 \leq K \leq 10^9$
• $1 \leq c_i \leq N$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$

$c_1$ $c_2$ $\dots$ $c_N$

Output

Print the answer.

Sample Input 1

4 1
1 1 2 1

Sample Output 1

3

Here is a comprehensive list of possible Takahashi's operations and permutations of people after each operation.

• Swap the positions of Person $1$ and Person $2$, resulting in a permutation $(2, 1, 3, 4)$.
• Swap the positions of Person $1$ and Person $4$, resulting in a permutation $(4, 2, 3, 1)$.
• Swap the positions of Person $2$ and Person $4$, resulting in a permutation $(1, 4, 3, 2)$.

Sample Input 2

3 3
1 1 2

Sample Output 2

1

Here is an example of a possible sequence of Takahashi's operations.

• In the $1$-st operation, he swaps the positions of Person $1$ and Person $3$, resulting in a permutation $(3, 2, 1)$.
  In the $2$-nd operation, he swaps the positions of Person $2$ and Person $3$, resulting in a permutation $(2, 3, 1)$.
  In the $3$-rd operation, he swaps the positions of Person $1$ and Person $3$, resulting in a permutation $(2, 1, 3)$.

Note that, during the sequence of operations, the color that the $i$-th person from the front is wearing does not necessarily coincide with $c_i$.

Sample Input 3

10 4
2 7 1 8 2 8 1 8 2 8

Sample Output 3

132

update @ 2024/3/10 10:36:32