Simple Speed

AtCoder

IDarc146_e

Time2000ms

Memory256MB

Difficulty—

You are given a sequence of $N$ positive integers: $A=(A_1,A_2,\dots,A_N)$. How many integer sequences $B$ consisting of integers between $1$ and $N$ (inclusive) satisfy all of the following conditions? Print the count modulo $998244353$. * For each integer $i$ such that $1 \le i \le N$, there are exactly $A_i$ occurrences of $i$ in $B$. * For each integer $i$ such that $1 \le i \le |B|-1$, it holds that $|B_i - B_{i+1}|=1$. ## Constraints * $1 \le N \le 2 \times 10^5$ * $1 \le A_i \le 2 \times 10^5$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\dots$ $A_N$ [samples]

Samples

Input #1

3
2 3 1

Output #1

6

$B$ can be the following six sequences.

*   $(1,2,1,2,3,2)$
*   $(1,2,3,2,1,2)$
*   $(2,1,2,1,2,3)$
*   $(2,1,2,3,2,1)$
*   $(2,3,2,1,2,1)$
*   $(3,2,1,2,1,2)$

Thus, the answer is $6$.

Input #2

1
200000

Output #2

0

There may be no sequence that satisfies the conditions.

Input #3

6
12100 31602 41387 41498 31863 12250

Output #3

750337372

API Response (JSON)

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  "problem": {
    "name": "Simple Speed",
    "description": {
      "content": "You are given a sequence of $N$ positive integers: $A=(A_1,A_2,\\dots,A_N)$. How many integer sequences $B$ consisting of integers between $1$ and $N$ (inclusive) satisfy all of the following condition",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc146_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a sequence of $N$ positive integers: $A=(A_1,A_2,\\dots,A_N)$.\nHow many integer sequences $B$ consisting of integers between $1$ and $N$ (inclusive) satisfy all of the following condition...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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