Strange Constraints

AtCoder

IDarc162_e

Time3000ms

Memory256MB

Difficulty—

You are given a sequence of length $N$ consisting of integers from $1$ to $N$, $A=(A_1,A_2,\ldots,A_N)$. Find the number, modulo $998244353$, of sequences of length $N$ consisting of integers from $1$ to $N$, $B=(B_1,B_2,\ldots,B_N)$, that satisfy the following conditions for all $i=1,2,\ldots,N$. * The number of occurrences of $i$ in $B$ is at most $A_i$. * The number of occurrences of $B_i$ in $B$ is at most $A_i$. ## Constraints * $1 \leq N \leq 500$ * $1 \leq A_i \leq N$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\ldots$ $A_N$ [samples]

Samples

Input #1

3
1 2 3

Output #1

10

The following $10$ sequences satisfy the conditions:

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

Input #2

4
4 4 4 4

Output #2

256

All sequences of length $4$ consisting of integers from $1$ to $4$ satisfy the conditions, and there are $4^4=256$ such sequences.

Input #3

5
1 1 1 1 1

Output #3

120

All permutations of $(1,2,3,4,5)$ satisfy the conditions, and there are $5!=120$ such sequences.

Input #4

14
6 5 14 3 6 7 3 11 11 2 3 7 8 10

Output #4

628377683

Be sure to print the number modulo $998244353$.

API Response (JSON)

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    "name": "Strange Constraints",
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      "content": "You are given a sequence of length $N$ consisting of integers from $1$ to $N$, $A=(A_1,A_2,\\ldots,A_N)$. Find the number, modulo $998244353$, of sequences of length $N$ consisting of integers from $1$",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
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    "difficulty": "None",
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    {
      "statement_type": "Markdown",
      "content": "You are given a sequence of length $N$ consisting of integers from $1$ to $N$, $A=(A_1,A_2,\\ldots,A_N)$.\nFind the number, modulo $998244353$, of sequences of length $N$ consisting of integers from $1$...",
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      "language": "English"
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