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Ex - Removing People

Score : $600$ points

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.

3
LLR

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 $3(=1+2)$.

10
RRRRRRLLRR

460301586