Shift and Inversions

AtCoder

IDabc190_f

Time2000ms

Memory256MB

Difficulty—

Given is a sequence $A = [a_0, a_1, a_2, \dots, a_{N-1}]$ that is a permutation of $0, 1, 2, \dots, N - 1$. For each $k = 0, 1, 2, \dots, N - 1$, find the inversion number of the sequence $B = [b_0, b_1, b_2, \dots, b_{N-1}]$ defined as $b_i = a_{i+k \bmod N}$. What is inversion number? The inversion number of a sequence $A = [a_0, a_1, a_2, \dots, a_{N-1}]$ is the number of pairs of indices $(i, j)$ such that $i < j$ and $a_i > a_j$. ## Constraints * All values in input are integers. * $2 ≤ N ≤ 3 \times 10^5$ * $a_0, a_1, a_2, \dots, a_{N-1}$ is a permutation of $0, 1, 2, \dots, N - 1$. ## Input Input is given from Standard Input in the following format: $N$ $a_0$ $a_1$ $a_2$ $\cdots$ $a_{N-1}$ [samples]

Samples

Input #1

4
0 1 2 3

Output #1

0
3
4
3

We have $A = [0, 1, 2, 3]$.
For $k = 0$, the inversion number of $B = [0, 1, 2, 3]$ is $0$.  
For $k = 1$, the inversion number of $B = [1, 2, 3, 0]$ is $3$.  
For $k = 2$, the inversion number of $B = [2, 3, 0, 1]$ is $4$.  
For $k = 3$, the inversion number of $B = [3, 0, 1, 2]$ is $3$.

Input #2

10
0 3 1 5 4 2 9 6 8 7

Output #2

API Response (JSON)

{
  "problem": {
    "name": "Shift and Inversions",
    "description": {
      "content": "Given is a sequence $A = [a_0, a_1, a_2, \\dots, a_{N-1}]$ that is a permutation of $0, 1, 2, \\dots, N - 1$.   For each $k = 0, 1, 2, \\dots, N - 1$, find the inversion number of the sequence $B = [b_0,",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc190_f"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given is a sequence $A = [a_0, a_1, a_2, \\dots, a_{N-1}]$ that is a permutation of $0, 1, 2, \\dots, N - 1$.  \nFor each $k = 0, 1, 2, \\dots, N - 1$, find the inversion number of the sequence $B = [b_0,...",
      "is_translate": false,
      "language": "English"
    }
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}

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