Problem:
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The Number of Inversions

For a given sequence $A = \{a_0, a_1, ... a_{n-1}\}$, the number of pairs $(i, j)$ where $a_i > a_j$ and $i < j$, is called the number of inversions. The number of inversions is equal to the number of swaps of Bubble Sort defined in the following program:

bubbleSort(A)
cnt = 0 // the number of inversions
for i = 0 to A.length-1
for j = A.length-1 downto i+1
if A[j] < A[j-1]
swap(A[j], A[j-1])
cnt++

return cnt

For the given sequence $A$, print the number of inversions of $A$. Note that you should not use the above program, which brings Time Limit Exceeded.

Input

In the first line, an integer $n$, the number of elements in $A$, is given. In the second line, the elements $a_i$ ($i = 0, 1, .. n-1$) are given separated by space characters.

output

Print the number of inversions in a line.

Constraints

$ 1 \leq n \leq 200,000$

$ 0 \leq a_i \leq 10^9$

$a_i$ are all different

Sample Input 1

5
3 5 2 1 4

Sample Output 1

6

Sample Input 2

3
3 1 2

Sample Output 2

2

Solution in Rust (judge verdict: Accepted)

/**
*  _           _                 __                            _   _ _   _                                 _                    _                  _
* | |         | |               / /                           | | (_) | (_)                               | |                  (_)                | |
* | |__   __ _| |_ ___   ___   / /__ ___  _ __ ___  _ __   ___| |_ _| |_ ___   _____ ______ _ __ _   _ ___| |_ ______ ___ _ __  _ _ __  _ __   ___| |_ ___
* | '_ \ / _` | __/ _ \ / _ \ / / __/ _ \| '_ ` _ \| '_ \ / _ \ __| | __| \ \ / / _ \______| '__| | | / __| __|______/ __| '_ \| | '_ \| '_ \ / _ \ __/ __|
* | | | | (_| | || (_) | (_) / / (_| (_) | | | | | | |_) |  __/ |_| | |_| |\ V /  __/      | |  | |_| \__ \ |_       \__ \ | | | | |_) | |_) |  __/ |_\__ \
* |_| |_|\__,_|\__\___/ \___/_/ \___\___/|_| |_| |_| .__/ \___|\__|_|\__|_| \_/ \___|      |_|   \__,_|___/\__|      |___/_| |_|_| .__/| .__/ \___|\__|___/
*                                                  | |                                                                           | |   | |
*                                                  |_|                                                                           |_|   |_|
*
* https://github.com/hatoo/competitive-rust-snippets
*/
#[allow(unused_imports)]
use std::cmp::{max, min, Ordering};
#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::iter::FromIterator;
#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};
pub mod util {
    use std::io::{stdin, stdout, BufWriter, StdoutLock};
    use std::str::FromStr;
    use std::fmt::Debug;
    #[allow(dead_code)]
    pub fn line() -> String {
        let mut line: String = String::new();
        stdin().read_line(&mut line).unwrap();
        line.trim().to_string()
    }
    #[allow(dead_code)]
    pub fn chars() -> Vec<char> {
        line().chars().collect()
    }
    #[allow(dead_code)]
    pub fn gets<T: FromStr>() -> Vec<T>
    where
        <T as FromStr>::Err: Debug,
    {
        let mut line: String = String::new();
        stdin().read_line(&mut line).unwrap();
        line.split_whitespace()
            .map(|t| t.parse().unwrap())
            .collect()
    }
    #[allow(dead_code)]
    pub fn with_bufwriter<F: FnOnce(BufWriter<StdoutLock>) -> ()>(f: F) {
        let out = stdout();
        let writer = BufWriter::new(out.lock());
        f(writer)
    }
}
#[allow(unused_macros)]
macro_rules ! get { ( $ t : ty ) => { { let mut line : String = String :: new ( ) ; stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; line . trim ( ) . parse ::<$ t > ( ) . unwrap ( ) } } ; ( $ ( $ t : ty ) ,* ) => { { let mut line : String = String :: new ( ) ; stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; let mut iter = line . split_whitespace ( ) ; ( $ ( iter . next ( ) . unwrap ( ) . parse ::<$ t > ( ) . unwrap ( ) , ) * ) } } ; ( $ t : ty ; $ n : expr ) => { ( 0 ..$ n ) . map ( | _ | get ! ( $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ ( $ t : ty ) ,*; $ n : expr ) => { ( 0 ..$ n ) . map ( | _ | get ! ( $ ( $ t ) ,* ) ) . collect ::< Vec < _ >> ( ) } ; ( $ t : ty ;; ) => { { let mut line : String = String :: new ( ) ; stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; line . split_whitespace ( ) . map ( | t | t . parse ::<$ t > ( ) . unwrap ( ) ) . collect ::< Vec < _ >> ( ) } } ; ( $ t : ty ;; $ n : expr ) => { ( 0 ..$ n ) . map ( | _ | get ! ( $ t ;; ) ) . collect ::< Vec < _ >> ( ) } ; }
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
const BIG_STACK_SIZE: bool = true;
#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}

fn merge_sort<T: Ord>(xs: &mut [T]) -> usize {
    if xs.len() <= 1 {
        return 0;
    }

    let len = xs.len();
    let mid = xs.len() / 2;

    let (sum, mut out) = {
        let (l, r) = xs.split_at_mut(mid);

        let a = merge_sort(l);
        let b = merge_sort(r);
        let mut out = Vec::with_capacity(len);
        unsafe {
            out.set_len(len);
        }
        let c = merge(l, r, &mut out);
        (a + b + c, out)
    };

    for (x, y) in xs.iter_mut().zip(out.iter_mut()) {
        std::mem::swap(x, y);
    }

    sum
}

fn merge<T: Ord>(l: &mut [T], r: &mut [T], out: &mut [T]) -> usize {
    let mut li = 0;
    let mut ri = 0;

    let mut count = 0;
    let mut i = 0;

    while li < l.len() && ri < r.len() {
        if l[li] <= r[ri] {
            std::mem::swap(&mut l[li], &mut out[i]);
            li += 1;
        } else {
            std::mem::swap(&mut r[ri], &mut out[i]);
            ri += 1;
            count += l.len() - li;
        }
        i += 1;
    }

    while li < l.len() {
        std::mem::swap(&mut l[li], &mut out[i]);
        li += 1;
        i += 1;
    }

    while ri < r.len() {
        std::mem::swap(&mut r[ri], &mut out[i]);
        ri += 1;
        i += 1;
    }

    count
}

#[allow(dead_code)]
/// Binary Indexed Tree of usize
pub struct BIT {
    buf: Vec<usize>,
}
#[allow(dead_code)]
impl BIT {
    pub fn new(n: usize) -> BIT {
        BIT {
            buf: vec![0; n + 1],
        }
    }
    pub fn sum(&self, i: usize) -> usize {
        let mut i = i;
        let mut s = 0;
        while i > 0 {
            s += self.buf[i];
            i &= i - 1;
        }
        s
    }
    pub fn add(&mut self, i: usize, x: usize) {
        let mut i = i as i64;
        while i < self.buf.len() as i64 {
            self.buf[i as usize] += x;
            i += i & -i;
        }
    }
}

fn solve() {
    let n = get!(usize);
    let xs = util::gets::<u64>();
    let mut cc = xs.clone();

    cc.sort();

    let mut bit = BIT::new(n);

    let mut ans = 0;
    for (i, x) in xs.into_iter().enumerate() {
        let x = cc.binary_search(&x).unwrap();
        ans += i - bit.sum(x + 1);
        bit.add(x + 1, 1);
    }
    println!("{}", ans);
    // println!("{}", merge_sort(&mut xs));
}