Line
Contest: Codeforces Round 817 (Div. 4)
Difficulty rating: 1100
Tags: greedy, sortings

There are $$$n$$$ people in a horizontal line, each looking either to the left or the right. Each person counts the number of people in the direction they are looking. The value of the line is the sum of each person's count.

For example, in the arrangement LRRLL, where L stands for a person looking left and R stands for a person looking right, the counts for each person are $$$[0, 3, 2, 3, 4]$$$, and the value is $$$0+3+2+3+4=12$$$.

You are given the initial arrangement of people in the line. For each $$$k$$$ from $$$1$$$ to $$$n$$$, determine the maximum value of the line if you can change the direction of at most $$$k$$$ people.

Input format:
The input consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains an integer $$$n$$$ ($$$1 \leq n \leq 2\cdot10^5$$$) — the length of the line.

The following line contains a string consisting of $$$n$$$ characters, each of which is either L or R, representing a person facing left or right, respectively — the description of the line.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2\cdot10^5$$$.

Please note that the answer for some test cases won't fit into 32-bit integer type, so you should use at least 64-bit integer type in your programming language (like long long for C++).

Output format:
For each test case, output $$$n$$$ space-separated non-negative integers — the maximum value of the line if you can change the direction of at most $$$k$$$ people for each $$$k$$$ from $$$1$$$ to $$$n$$$, inclusive.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
In the first test case:

- $$$k=1$$$: change the direction of $$$1$$$ person to make the line RLR. The total value is $$$2+1+0=3$$$.
- $$$k=2$$$: change the direction of $$$2$$$ people to make the line RLL. The total value is $$$2+1+2=5$$$.
- $$$k=3$$$: change the direction of $$$2$$$ people to make the line RLL. The total value is $$$2+1+2=5$$$. Note that you have to change the direction of at most $$$k$$$ people.

In the second test case, it is optimal to only change the direction of the first person for all $$$k$$$ from $$$1$$$ to $$$5$$$ (that is, make the line RRRLL).

Example input:
6
3
LLR
5
LRRLL
1
L
12
LRRRLLLRLLRL
10
LLLLLRRRRR
9
LRLRLRLRL

Example output:
3 5 5 
16 16 16 16 16 
0 
86 95 98 101 102 102 102 102 102 102 102 102 
29 38 45 52 57 62 65 68 69 70 
44 50 54 56 56 56 56 56 56

Editorial:
For each person, let's calculate how much the value will change if they turn around. For example, in the line LRRLL, if the $$$i$$$-th person turns around, then the value of the line will change by $$$+4$$$, $$$-2$$$, $$$0$$$, $$$-2$$$, $$$-4$$$, respectively. (For instance, if the second person turns around, they see $$$3$$$ people before and $$$1$$$ person after, so the value of the line changes by $$$-2$$$ if they turn around.)
Now note that if a person turns around, it doesn't affect anyone else's value. So the solution is a greedy one: let's sort the array of values in increasing order. Afterwards, we should go from the left to the right, and see if the value will increase if this person turns around; if it does, we should add it to the current total and continue.
The time complexity of this solution is $$$\mathcal{O}(n \log n)$$$ per testcase.