Cards

AtCoder

IDabc247_f

Time2000ms

Memory256MB

Difficulty—

There are $N$ cards numbered $1,\ldots,N$. Card $i$ has $P_i$ written on the front and $Q_i$ written on the back. Here, $P=(P_1,\ldots,P_N)$ and $Q=(Q_1,\ldots,Q_N)$ are permutations of $(1, 2, \dots, N)$. How many ways are there to choose some of the $N$ cards such that the following condition is satisfied? Find the count modulo $998244353$. Condition: every number $1,2,\ldots,N$ is written on at least one of the chosen cards. ## Constraints * $1 \leq N \leq 2\times 10^5$ * $1 \leq P_i,Q_i \leq N$ * $P$ and $Q$ are permutations of $(1, 2, \dots, N)$. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $P_1$ $P_2$ $\ldots$ $P_N$ $Q_1$ $Q_2$ $\ldots$ $Q_N$ [samples]

Samples

Input #1

3
1 2 3
2 1 3

Output #1

3

For example, if you choose Cards $1$ and $3$, then $1$ is written on the front of Card $1$, $2$ on the back of Card $1$, and $3$ on the front of Card $3$, so this combination satisfies the condition.
There are $3$ ways to choose cards satisfying the condition: ${1,3},{2,3},{1,2,3}$.

Input #2

5
2 3 5 4 1
4 2 1 3 5

Output #2

Input #3

8
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Cards",
    "description": {
      "content": "There are $N$ cards numbered $1,\\ldots,N$. Card $i$ has $P_i$ written on the front and $Q_i$ written on the back.   Here, $P=(P_1,\\ldots,P_N)$ and $Q=(Q_1,\\ldots,Q_N)$ are permutations of $(1, 2, \\dot",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc247_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ cards numbered $1,\\ldots,N$. Card $i$ has $P_i$ written on the front and $Q_i$ written on the back.  \nHere, $P=(P_1,\\ldots,P_N)$ and $Q=(Q_1,\\ldots,Q_N)$ are permutations of $(1, 2, \\dot...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments