 Problem 1. Haircut

Contest has ended. Tired of his stubborn cowlick, Farmer John decides to get a haircut. He has $N$ ($1\le N\le 10^5$) strands of hair arranged in a line, and strand $i$ is initially $A_i$ micrometers long ($0\le A_i\le N$). 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 $A_i > A_j$.

For each of $j=0,1,\ldots,N-1$, 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 $10^5$ hairs!)

INPUT FORMAT (file haircut.in):

The first line contains $N$.

The second line contains $A_1,A_2,\ldots,A_N.$

OUTPUT FORMAT (file haircut.out):

For each of $j=0,1,\ldots,N-1$, 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++).

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: $A_1>A_2,\,A_1>A_5,\,A_2>A_5,\,A_3>A_5,$ and $A_4>A_5$.

SCORING:

• Test case 2 satisfies $N\le 100.$
• Test cases 3-5 satisfy $N\le 5000.$
• Test cases 6-13 satisfy no additional constraints.

Problem credits: Dhruv Rohatgi

Contest has ended. No further submissions allowed.