Problem:
Score : 300 points

Problem Statement

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}

Output

Print the inversion number of B, modulo 10^9 + 7.

Sample Input 1

2 2
2 1

Sample 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.

Sample Input 2

3 5
1 1 1

Sample Output 2

0

A may contain multiple occurrences of the same number.

Sample Input 3

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

Sample Output 3

185297239

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

Solution in Rust (1.15.1) (judge verdict: Accepted)

#![allow(unused_imports)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), std::io::stderr());
            writeln!(err, "{} = {:?}", e, $e).unwrap()
        })*
    };
}

fn main() {
    let v = read_vec::<i64>();
    let (n, k) = (v[0] as usize, v[1]);
    let a = read_vec::<usize>();
    if n == 1 {
        println!("0");
        return;
    }
    let aidx = argsort(&a);
    let aidx = aidx.into_iter().map(|x| x + 1).collect::<Vec<_>>();
    let inv = count_inversion(&aidx);
    // let inv = count_inversion(&a);
    let big_int = 1000_000_000i64 + 7;

    let mut ans = inv * k;
    ans %= big_int;

    let mut additional = ((k % big_int) * ((k - 1) % big_int) / 2) % big_int;
    debug!(additional);

    let mut a = a.iter().map(|&x| -1 * x as i64).collect::<Vec<_>>();
    a.sort();
    for i in 0..n {
        let residual = (a.len() - a.upper_bound(&a[i])) as i64;
        let add = (residual * additional) % big_int;
        ans += add;
        ans %= big_int;
    }

    println!("{}", ans);
}

fn count_inversion(v: &Vec<usize>) -> i64 {
    let mut ans = 0;
    let mut bt = BinaryIndexTree::new(v.len());
    for i in 0..v.len() {
        ans += i as i64 - bt.sum(v[i]);
        bt.add(v[i], 1);
    }
    ans
}

struct BinaryIndexTree {
    bit: Vec<i64>,
    n: usize,
}

impl BinaryIndexTree {
    fn new(n: usize) -> BinaryIndexTree {
        BinaryIndexTree {
            bit: vec![0; n + 1],
            n: n,
        }
    }

    fn sum(&self, i: usize) -> i64 {
        let mut i = i;
        let mut s = 0i64;
        while i > 0 {
            s += self.bit[i];
            i -= (i as i64 & -(i as i64)) as usize;
        }
        s
    }

    fn add(&mut self, i: usize, x: i64) {
        let mut i = i;
        while i <= self.n {
            self.bit[i as usize] += x;
            i += (i as i64 & -(i as i64)) as usize;
        }
    }
}

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}

pub trait BinarySearch<T> {
    fn lower_bound(&self, x: &T) -> usize;
    fn upper_bound(&self, x: &T) -> usize;
}

impl<T: Ord> BinarySearch<T> for [T] {
    fn lower_bound(&self, x: &T) -> usize {
        let mut low = 0;
        let mut high = self.len();

        while low != high {
            let mid = (low + high) / 2;
            match self[mid].cmp(x) {
                Ordering::Less => {
                    low = mid + 1;
                }
                Ordering::Equal | Ordering::Greater => {
                    high = mid;
                }
            }
        }
        low
    }

    fn upper_bound(&self, x: &T) -> usize {
        let mut low = 0;
        let mut high = self.len();

        while low != high {
            let mid = (low + high) / 2;
            match self[mid].cmp(x) {
                Ordering::Less | Ordering::Equal => {
                    low = mid + 1;
                }
                Ordering::Greater => {
                    high = mid;
                }
            }
        }
        low
    }
}

fn argsort<T: std::cmp::Ord + std::marker::Copy>(v: &Vec<T>) -> Vec<usize> {
    let mut sort_v = Vec::new();
    for i in 0..v.len() {
        sort_v.push((v[i], i));
    }
    sort_v.sort();
    sort_v.iter().map(|x| x.1).collect::<Vec<usize>>()
}