Erase and Insert

AtCoder

IDagc065_b

Time2000ms

Memory256MB

Difficulty—

You are given a permutation $P$ of $(1,2,\dots,N)$. There is also a permutation $Q=(1,2,\dots,N)$ of $(1,2,\dots,N)$. Perform the following operation on $Q$ for $i=1,2,\dots,N$ in this order. * Remove $i$ from $Q$ and insert $i$ into any one position in $Q$. Find the number, modulo $(10^9+7)$, of ways to perform $N$ operations such that $P$ and $Q$ are equal after all operations. ## Constraints * $1 \le N \le 5000$ * $P$ is a permutation of $(1,2,\dots,N)$. ## Input The input is given from Standard Input in the following format: $N$ $P_1$ $P_2$ $\dots$ $P_N$ [samples]

Samples

Input #1

3
1 2 3

Output #1

5

For example, the following operations result in $Q = (1,2,3)$.

*   Remove $1$ from $Q=(1,2,3)$ and insert it between $2$ and $3$, making $Q = (2,1,3)$.
*   Remove $2$ from $Q=(2,1,3)$ and insert it at the end of $Q$, making $Q = (1,3,2)$.
*   Remove $3$ from $Q=(1,3,2)$ and insert it at the end of $Q$, making $Q = (1,2,3)$.

Including this example, five ways to perform operations result in $Q=(1,2,3)$.

Input #2

4
2 4 1 3

Output #2

Input #3

15
7 5 14 10 4 2 3 6 8 11 12 1 15 13 9

Output #3

Input #4

30
15 19 13 11 22 27 21 25 1 12 30 28 16 26 10 14 20 2 5 7 23 4 17 6 29 3 18 9 8 24

Output #4

33525150

API Response (JSON)

{
  "problem": {
    "name": "Erase and Insert",
    "description": {
      "content": "You are given a permutation $P$ of $(1,2,\\dots,N)$. There is also a permutation $Q=(1,2,\\dots,N)$ of $(1,2,\\dots,N)$. Perform the following operation on $Q$ for $i=1,2,\\dots,N$ in this order. *   Rem",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc065_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a permutation $P$ of $(1,2,\\dots,N)$. There is also a permutation $Q=(1,2,\\dots,N)$ of $(1,2,\\dots,N)$.\nPerform the following operation on $Q$ for $i=1,2,\\dots,N$ in this order.\n\n*   Rem...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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