Kleene Inversion

AtCoder

IDjsc2019_qual_b

Time2000ms

Memory256MB

Difficulty—

We have a sequence of $N$ integers $A~=~A_0,~A_1,~...,~A_{N - 1}$. Let $B$ be a sequence of $K \times N$ integers obtained by concatenating $K$ copies of $A$. For example, if $A~=~1,~3,~2$ and $K~=~2$, $B~=~1,~3,~2,~1,~3,~2$. Find the inversion number of $B$, modulo $10^9 + 7$. Here the inversion number of $B$ is defined as the number of ordered pairs of integers $(i,~j)~(0 \leq i < j \leq K \times N - 1)$ such that $B_i > B_j$. ## Constraints * All values in input are integers. * $1 \leq N \leq 2000$ * $1 \leq K \leq 10^9$ * $1 \leq A_i \leq 2000$ ## Input Input is given from Standard Input in the following format: $N$ $K$ $A_0$ $A_1$ $...$ $A_{N - 1}$ [samples]

Samples

Input #1

2 2
2 1

Output #1

3

In this case, $B~=~2,~1,~2,~1$. We have:

*   $B_0 > B_1$
*   $B_0 > B_3$
*   $B_2 > B_3$

Thus, the inversion number of $B$ is $3$.

Input #2

3 5
1 1 1

Output #2

0

$A$ may contain multiple occurrences of the same number.

Input #3

10 998244353
10 9 8 7 5 6 3 4 2 1

Output #3

185297239

Be sure to print the output modulo $10^9 + 7$.

API Response (JSON)

{
  "problem": {
    "name": "Kleene Inversion",
    "description": {
      "content": "We have a sequence of $N$ integers $A~=~A_0,~A_1,~...,~A_{N - 1}$. Let $B$ be a sequence of $K \\times N$ integers obtained by concatenating $K$ copies of $A$. For example, if $A~=~1,~3,~2$ and $K~=~2$",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "jsc2019_qual_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a sequence of $N$ integers $A~=~A_0,~A_1,~...,~A_{N - 1}$.\nLet $B$ be a sequence of $K \\times N$ integers obtained by concatenating $K$ copies of $A$. For example, if $A~=~1,~3,~2$ and $K~=~2$...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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