Problem 3. Cow Checkups

USACO 2025 January Contest, Bronze

Note: We suggest using a language other than Python to earn full credit on this problem.

Farmer John's $N$ ($1 \leq N \leq 7500$) cows are standing in a line, with cow $1$ at the front of the line and cow $N$ at the back of the line. FJ's cows also come in many different species. He denotes each species with an integer from $1$ to $N$. The $i$'th cow from the front of the line is of species $a_i$ ($1 \leq a_i \leq N$).

FJ is taking his cows to a checkup at a local bovine hospital. However, the bovine veterinarian is very picky and wants to perform a checkup on the $i$'th cow in the line, only if it is of species $b_i$ ($1 \leq b_i \leq N$).

FJ is lazy and does not want to completely reorder his cows. He will perform the following operation

exactly once

.

• Select two integers $l$ and $r$ such that $1 \leq l \le r \leq N$. Reverse the order of the cows that are between the $l$-th cow and the $r$-th cow in the line, inclusive.

FJ wants to measure how effective this approach is. For each $c=0 \ldots N$, help FJ find the number of distinct operations ($l,r$) that result in exactly $c$ cows being checked. Two operations ($l_1,r_1$) and ($l_2,r_2$) are different if $l_1 \neq l_2$ or $r_1 \neq r_2$.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains an integer $N$.

The second line contains $a_1, a_2, \ldots, a_N$.

The third line contains $b_1, b_2, \ldots, b_N$.

OUTPUT FORMAT (print output to the terminal / stdout):

Output $N+1$ lines with the $i$-th line containing the number of distinct operations ($l,r$) that result in $i-1$ cows being checked.

SAMPLE INPUT:

3
1 3 2
3 2 1

SAMPLE OUTPUT:

3
3
0
0

If FJ chooses ($l=1,r=1$), ($l=2,r=2$), or ($l=3,r=3$) then no cows will be checked. Note that those operations do not modify any of the cows' locations.

The following operations result in one cow being checked.

• $l=1,r=2$: FJ reverses the order of the first and second cows so the species of each cow in the new lineup will be $[3,1,2]$. The first cow will be checked.
• $l=2,r=3$: FJ reverses the order of the second and third cows so the species of each cow in the new lineup will be $[1,2,3]$. The second cow will be checked.
• $l=1,r=3$: FJ reverses the order of the first, second, and third cows so the species of each cow in the new lineup will be $[2,3,1]$. The third cow will be checked.

SAMPLE INPUT:

3
1 2 3
1 2 3

SAMPLE OUTPUT:

0
3
0
3

The three possible operations that cause $3$ cows to be checked are ($l=1,r=1$), ($l=2,r=2$), and ($l=3,r=3$).

SAMPLE INPUT:

7
1 3 2 2 1 3 2
3 2 2 1 2 3 1

SAMPLE OUTPUT:

0
6
14
6
2
0
0
0

The two possible operations that cause $4$ cows to be checked are ($l=4,r=5$) and ($l=5,r=7$).

SCORING: Inputs 4-6: $N\le 100$Inputs 7-13: No additional constraints

Problem credits: Chongtian Ma and Haokai Ma