Random LIS

AtCoder

IDarc104_e

Time2000ms

Memory256MB

Difficulty—

Given is an integer sequence of length $N$: $A_1, A_2, \cdots, A_N$. An integer sequence $X$, which is also of length $N$, will be chosen randomly by independently choosing $X_i$ from a uniform distribution on the integers $1, 2, \ldots, A_i$ for each $i (1 \leq i \leq N)$. Compute the expected value of the length of the longest increasing subsequence of this sequence $X$, modulo $1000000007$. More formally, under the constraints of the problem, we can prove that the expected value can be represented as a rational number, that is, an irreducible fraction $\frac{P}{Q}$, and there uniquely exists an integer $R$ such that $R \times Q \equiv P \pmod {1000000007}$ and $0 \leq R < 1000000007$, so print this integer $R$. ## Constraints * $1 \leq N \leq 6$ * $1 \leq A_i \leq 10^9$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\cdots$ $A_N$ [samples] ## Notes A subsequence of a sequence $X$ is a sequence obtained by extracting some of the elements of $X$ and arrange them without changing the order. The longest increasing subsequence of a sequence $X$ is the sequence of the greatest length among the strictly increasing subsequences of $X$.

Samples

Input #1

3
1 2 3

Output #1

2

$X$ becomes one of the following, with probability $1/6$ for each:

*   $X = (1,1,1)$, for which the length of the longest increasing subsequence is $1$;
*   $X = (1,1,2)$, for which the length of the longest increasing subsequence is $2$;
*   $X = (1,1,3)$, for which the length of the longest increasing subsequence is $2$;
*   $X = (1,2,1)$, for which the length of the longest increasing subsequence is $2$;
*   $X = (1,2,2)$, for which the length of the longest increasing subsequence is $2$;
*   $X = (1,2,3)$, for which the length of the longest increasing subsequence is $3$.

Thus, the expected value of the length of the longest increasing subsequence is $(1 + 2 + 2 + 2 + 2 + 3) / 6 \equiv 2 \pmod {1000000007}$.

Input #2

3
2 1 2

Output #2

500000005

$X$ becomes one of the following, with probability $1/4$ for each:

*   $X = (1,1,1)$, for which the length of the longest increasing subsequence is $1$;
*   $X = (1,1,2)$, for which the length of the longest increasing subsequence is $2$;
*   $X = (2,1,1)$, for which the length of the longest increasing subsequence is $1$;
*   $X = (2,1,2)$, for which the length of the longest increasing subsequence is $2$.

Thus, the expected value of the length of the longest increasing subsequence is $(1 + 2 + 1 + 2) / 4 = 3 / 2$. Since $2 \times 500000005 \equiv 3 \pmod {1000000007}$, we should print $500000005$.

Input #3

6
8 6 5 10 9 14

Output #3

959563502

API Response (JSON)

{
  "problem": {
    "name": "Random LIS",
    "description": {
      "content": "Given is an integer sequence of length $N$: $A_1, A_2, \\cdots, A_N$. An integer sequence $X$, which is also of length $N$, will be chosen randomly by independently choosing $X_i$ from a uniform distri",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc104_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given is an integer sequence of length $N$: $A_1, A_2, \\cdots, A_N$.\nAn integer sequence $X$, which is also of length $N$, will be chosen randomly by independently choosing $X_i$ from a uniform distri...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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