102265: [AtCoder]ABC226 F - Score of Permutations

Problem Statement

For a permutation $P = (p_1, p_2, \dots, p_N)$ of $(1,2, \dots, N)$, let us define the score $S(P)$ of $P$ as follows.

• There are $N$ people, numbered $1,2,\dots,N$. Additionally, Snuke is there. Initially, Person $i$ $(1 \leq i \leq N)$ has Ball $i$.
  Each time Snuke screams, every Person $i$ such that $i \neq p_i$ gives their ball to Person $p_i$ simultaneously.
  If, after screaming at least once, every Person $i$ has Ball $i$, Snuke stops screaming.
  The score is the number of times Snuke screams until he stops. Here, it is guaranteed that the score will be a finite value.

There are $N!$ permutations $P$ of $(1,2, \dots, N)$. Find the sum, modulo $998244353$, of the scores of those permutations, each raised to the $K$-th power.

• Formally, let $S_N$ be the set of the permutations of $(1,2, \dots, N)$. Compute the following: $\displaystyle \left(\sum_{P \in S_N} S(P)^K \right) \bmod {998244353}$.

Constraints

• $2 \leq N \leq 50$
• $1 \leq K \leq 10^4$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$

Output

Print $\displaystyle \left(\sum_{P \in S_N} S(P)^K \right) \bmod {998244353}$.

Sample Input 1

2 2

Sample Output 1

5

When $N = 2$, there are two possible permutations $P$: $(1,2),(2,1)$.

The score of the permutation $(1,2)$ is found as follows.

• Initially, Person $1$ has Ball $1$, and Person $2$ has Ball $2$.
  After Snuke's first scream, Person $1$ has Ball $1$, and Person $2$ has Ball $2$.
  Here, every Person $i$ has Ball $i$, so he stops screaming.
  Thus, the score is $1$.

The score of the permutation $(2,1)$ is found as follows.

• Initially, Person $1$ has Ball $1$, and Person $2$ has Ball $2$.
  After Snuke's first scream, Person $1$ has Ball $2$, and Person $2$ has Ball $1$.
  After Snuke's second scream, Person $1$ has Ball $1$, and Person $2$ has Ball $2$.
  Here, every Person $i$ has Ball $i$, so he stops screaming.
  Thus, the score is $2$.

Therefore, the answer in this case is $1^2 + 2^2 = 5$.

Sample Input 2

3 3

Sample Output 2

79

All permutations and their scores are listed below.

• $(1,2,3)$: The score is $1$.
• $(1,3,2)$: The score is $2$.
• $(2,1,3)$: The score is $2$.
• $(2,3,1)$: The score is $3$.
• $(3,1,2)$: The score is $3$.
• $(3,2,1)$: The score is $2$.

Thus, we should print $1^3 + 2^3 + 2^3 + 3^3 + 3^3 + 2^3 = 79$.

Sample Input 3

50 10000

Sample Output 3

77436607

问题描述

对于排列$P = (p_1, p_2, \dots, p_N)$，定义$P$的分数$S(P)$如下。

• 有$N$个人，编号为$1,2,\dots,N$。此外，Snuke也在场。最初，第$i$个人（$1 \leq i \leq N$）拥有Ball $i$。
  每次Snuke尖叫时，所有编号不为$p_i$的第$i$个人同时将他们的球交给第$p_i$个人。
  如果，在至少尖叫一次之后，每个人都拥有Ball $i$，则Snuke停止尖叫。
  分数为Snuke停止尖叫之前尖叫的次数。这里，保证分数将是一个有限值。

有$N!$个排列$P$为$(1,2, \dots, N)$。求这些排列的分数的和，模$998244353$，每个排列的分数都提高到$K$次方。

• 形式上，让$S_N$为$(1,2, \dots, N)$的排列的集合。计算以下值： $\displaystyle \left(\sum_{P \in S_N} S(P)^K \right) \bmod {998244353}$。

约束条件

• $2 \leq N \leq 50$
• $1 \leq K \leq 10^4$
• 输入中的所有值都是整数。

输入

输入从标准输入中给出，格式如下：

$N$ $K$

输出

打印$\displaystyle \left(\sum_{P \in S_N} S(P)^K \right) \bmod {998244353}$。

样例输入1

2 2

样例输出1

5

当$N = 2$时，有两种可能的排列$P$：$(1,2),(2,1)$。

排列$(1,2)$的分数如下找到。

• 最初，第$1$个人拥有Ball $1$，第$2$个人拥有Ball $2$。
  在Snuke的第一次尖叫后，第$1$个人拥有Ball $1$，第$2$个人拥有Ball $2$。
  这里，每个人都拥有Ball $i$，所以他停止尖叫。
  因此，分数为$1$。

排列$(2,1)$的分数如下找到。

• 最初，第$1$个人拥有Ball $1$，第$2$个人拥有Ball $2$。
  在Snuke的第一次尖叫后，第$1$个人拥有Ball $2$，第$2$个人拥有Ball $1$。
  在Snuke的第二次尖叫后，第$1$个人拥有Ball $1$，第$2$个人拥有Ball $2$。
  这里，每个人都拥有Ball $i$，所以他停止尖叫。
  因此，分数为$2$。

因此，在这种情况下，答案为$1^2 + 2^2 = 5$。

样例输入2

3 3

样例输出2

79

所有排列及其分数如下列出。

• $(1,2,3)$：分数为$1$。
• $(1,3,2)$：分数为$2$。
• $(2,1,3)$：分数为$2$。
• $(2,3,1)$：分数为$3$。
• $(3,1,2)$：分数为$3$。
• $(3,2,1)$：分数为$2$。

因此，我们应该打印$1^3 + 2^3 + 2^3 + 3^3 + 3^3 + 2^3 = 79$。

样例输入3

50 10000

样例输出3

77436607