Ex - Removing People

AtCoder

IDabc238_h

Time3000ms

Memory256MB

Difficulty—

$N$ 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). ## 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: $N$ $S$ [samples] ## 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 $R$.

Samples

Input #1

3
LLR

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 $3(=1+2)$.

Input #2

10
RRRRRRLLRR

Output #2

460301586

API Response (JSON)

{
  "problem": {
    "name": "Ex - Removing People",
    "description": {
      "content": "$N$ 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 $N$.",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc238_h"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "$N$ 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 $N$....",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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