Spoon Taking Problem

AtCoder

IDarc175_a

Time2000ms

Memory256MB

Difficulty—

$N$ people are sitting around a round table, and numbered $1$ to $N$ in a counterclockwise order. Each person has a dominant hand: left or right. There are $N$ spoons numbered $1$ to $N$ on the round table, with one spoon placed between each pair of adjacent people. For each $1 \leq i \leq N$, to the left and right of person $i$, there are spoons $i$ and $(i+1)$, respectively. Here, spoon $(N+1)$ refers to spoon $1$. Below is a diagram for $N = 4$. ![image](https://img.atcoder.jp/arc175/b86aef99039c82389bf15f8df725a4c5.png) You are given a permutation $(P_1, \dots, P_N)$ of $(1, \dots, N)$. In the order $i=1,\dots,N$, person $P_i$ will act as follows: * If there is a spoon remaining on left or right side, they will take one of them. * If there are spoons remaining on both sides, they will take the spoon on the side of their dominant hand. * Otherwise, they do nothing. You are also given a string $S$ of length $N$ consisting of `L`, `R`, and `?`. Among the $2^N$ possible combinations of dominant hands, find how many satisfy all of the following conditions, modulo $998244353$: * If the $i$\-th character of $S$ is `L`, person $i$ is left-handed. * If the $i$\-th character of $S$ is `R`, person $i$ is right-handed. * When everyone has finished acting, everyone has taken a spoon. ## Constraints * All input values are integers. * $2 \leq N \leq 2 \times 10^5$ * $(P_1, \dots, P_N)$ is a permutation of $(1, \dots, N)$. * $S$ is a string of length $N$ consisting of `L`, `R`, and `?`. ## Input The input is given from Standard Input in the following format: $N$ $P_1$ $\dots$ $P_N$ $S$ [samples]

Samples

Input #1

3
1 2 3
L??

Output #1

2

When persons $1$, $2$, and $3$ are left-handed, left-handed, and right-handed, respectively, the actions are performed as follows:

*   Person $1$ starts acting. There are spoons on both sides, so person $1$ takes spoon $1$ on the left side, which is the same as their dominant hand.
*   Person $2$ starts acting. There are spoons on both sides, so person $2$ takes spoon $2$ on the left side, which is the same as their dominant hand.
*   Person $3$ starts acting. There is no spoon on the right side, but spoon $3$ is remaining on the left side, so they take spoon $3$. Everyone has finished acting and taken a spoon.

This combination of dominant hands satisfies the conditions. Besides, the conditions are also satisfied when persons $1, 2, 3$ are all left-handed.

Input #2

3
1 3 2
R?L

Output #2

0

No combinations of dominant hands satisfy the conditions.

Input #3

12
6 2 9 3 1 4 11 5 12 10 7 8
????????????

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Spoon Taking Problem",
    "description": {
      "content": "$N$ people are sitting around a round table, and numbered $1$ to $N$ in a counterclockwise order. Each person has a dominant hand: left or right. There are $N$ spoons numbered $1$ to $N$ on the round ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc175_a"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "$N$ people are sitting around a round table, and numbered $1$ to $N$ in a counterclockwise order. Each person has a dominant hand: left or right.\nThere are $N$ spoons numbered $1$ to $N$ on the round ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments