Score : $600$ points

问题描述

设有$N$个人，编号为$1$至$N$，他们按照顺时针顺序站成一个圆圈：第$1$个人、第$2$个人、$\cdots$、第$N$个人。每个人面向的方向由长度为$N$的字符串$S$给出。对于每个$i$ $(1 \leq i \leq N)$，如果$S_i =$ L，则第$i$个人面向逆时针方向；如果$S_i =$ R，则第$i$个人面向顺时针方向。

接下来将重复进行$N-1$次以下操作：

• 等概率地选择一个剩余的人，然后从圆圈中移除从该人选出的人看到的最近的人。这次操作的成本等于选出的人到被移除的人的距离。

这里，从第$i$个人到第$j$个人$(i \neq j)$的距离定义如下：

1. 当第$i$个人面向顺时针方向时：
   • 如果$i < j$，距离为$j-i$；
   • 如果$i > j$，距离为$j-i+N$。
2. 当第$i$个人面向逆时针方向时：
   • 如果$i < j$，距离为$i-j+N$；
   • 如果$i > j$，距离为$i-j$。

求总成本的期望值对$998244353$取模的结果（参见注释）。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

$N$ people numbered $1$ to $N$ are standing in a circle, in the clockwise order of Person $1$, Person $2$, $\cdots$, Person $N$. The direction each person faces is given by a string $S$ of length $N$. For each $i$ $(1 \leq i \leq N)$, Person $i$ is facing in the counter-clockwise direction if $S_i =$ L, and clockwise direction if $S_i =$ R.

The following operation will be repeated $N-1$ times.

• Choose one of the remaining people with equal probability, and remove from the circle the nearest person seen from the chosen person. This incurs a cost equal to the distance from the chosen person to the removed person.

Here, the distance from Person $i$ to Person $j$ $(i \neq j)$ is defined as follows.

1. When Person $i$ is facing in the clockwise direction:
   • $j-i$ if $i \lt j$;
   • $j-i+N$ if $i \gt j$.
2. When Person $i$ is facing in the counter-clockwise direction:
   • $i-j+N$ if $i \lt j$;
   • $i-j$ if $i \gt j$.

Find the expected value of the total cost incurred, modulo $998244353$ (see Notes).

Notes

It can be proved that the sought expected value is always a rational number. Additionally, under the Constraints of this problem, when that value is expressed as $\frac{P}{Q}$ using two coprime integers $P$ and $Q$, there is a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Find this $R$.

Constraints

• $2 \leq N \leq 300$
• $N$ is an integer.
• $S$ is a string of length $N$ consisting of L and R.

Input

Input is given from Standard Input in the following format:

$N$

$S$

Output

Print the answer.

Sample Input 1

3
LLR

Sample Output 1

831870297

The sought expected value is $\frac{17}{6}$. We have $831870297 \times 6 \equiv 17\pmod{998244353}$, so $831870297$ should be printed.

For your reference, here is one possible scenario.

1. Person $2$ is chosen. The nearest person seen from Person $2$ remaining in the circle is Person $1$, who gets removed from the circle.
2. Person $2$ is chosen again. The nearest person seen from Person $2$ remaining in the circle is Person $3$, who gets removed from the circle.

In this case, the total cost incurred is $3(=1+2)$.

Sample Input 2

10
RRRRRRLLRR

Sample Output 2

460301586

update @ 2024/3/10 10:18:36