Processing math: 100%

USACO 2020 US Open Contest, Gold

Problem 1. Haircut


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Tired of his stubborn cowlick, Farmer John decides to get a haircut. He has N (1N105) strands of hair arranged in a line, and strand i is initially Ai micrometers long (0AiN). Ideally, he wants his hair to be monotonically increasing in length, so he defines the "badness" of his hair as the number of inversions: pairs (i,j) such that i<j and Ai>Aj.

For each of j=0,1,,N1, FJ would like to know the badness of his hair if all strands with length greater than j are decreased to length exactly j.

(Fun fact: the average human head does indeed have about 105 hairs!)

INPUT FORMAT (file haircut.in):

The first line contains N.

The second line contains A1,A2,,AN.

OUTPUT FORMAT (file haircut.out):

For each of j=0,1,,N1, output the badness of FJ's hair on a new line.

Note that the large size of integers involved in this problem may require the use of 64-bit integer data types (e.g., a "long long" in C/C++).

SAMPLE INPUT:

5
5 2 3 3 0

SAMPLE OUTPUT:

0
4
4
5
7

The fourth line of output describes the number of inversions when FJ's hairs are decreased to length 3. Then A=[3,2,3,3,0] has five inversions: A1>A2,A1>A5,A2>A5,A3>A5, and A4>A5.

SCORING:

  • Test case 2 satisfies N100.
  • Test cases 3-5 satisfy N5000.
  • Test cases 6-13 satisfy no additional constraints.

Problem credits: Dhruv Rohatgi

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