USACO 2024 US Open Contest, Gold

Problem 3. Smaller Averages


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Bessie has two arrays of length $N$ ($1 \le N \le 500$). The $i$-th element of the first array is $a_i$ ($1 \le a_i \le 10^6$) and the $i$-th element of the second array is $b_i$ ($1 \le b_i \le 10^6$).

Bessie wants to split both arrays into non-empty subarrays such that the following is true.

  1. Every element belongs in exactly 1 subarray.
  2. Both arrays are split into the same number of subarrays. Let the number of subarrays the first and second array are split into be $k$ (i.e. the first array is split into exactly $k$ subarrays and the second array is split into exactly $k$ subarrays).
  3. For all $1 \le i \le k$, the average of the $i$-th subarray on the left of the first array is less than or equal to the average of the $i$-th subarray on the left of the second array.

Count how many ways she can split both arrays into non-empty subarrays while satisfying the constraints modulo $10^9+7$. Two ways are considered different if the number of subarrays are different or if some element belongs in a different subarray.

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

The first line contains $N$.

The next line contains $a_1,a_2,...,a_N$.

The next line contains $b_1,b_2,...,b_N$.

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

Output the number of ways she can split both arrays into non-empty subarrays while satisfying the constraints modulo $10^9+7$.

SAMPLE INPUT:

2
1 2
2 2

SAMPLE OUTPUT:

2
The two valid ways are:
  1. Split the first array into $[1],[2]$ and the second array into $[2],[2]$.
  2. Split the first array into $[1,2]$ and the second array into $[2,2]$.

SAMPLE INPUT:

3
1 3 2
2 2 2

SAMPLE OUTPUT:

3
The three valid ways are:
  1. Split the first array into $[1,3],[2]$ and the second array into $[2,2],[2]$.
  2. Split the first array into $[1,3],[2]$ and the second array into $[2],[2,2]$.
  3. Split the first array into $[1,3,2]$ and the second array into $[2,2,2]$.

SAMPLE INPUT:

5
2 5 1 3 2
2 1 5 2 2

SAMPLE OUTPUT:

1
The only valid way is to split the first array into $[2],[5,1,3],[2]$ and the second array into $[2],[1,5],[2,2]$.

SAMPLE INPUT:

7
3 5 2 3 4 4 1
5 3 5 3 3 4 1

SAMPLE OUTPUT:

140

SCORING:

  • Inputs 5-6: $N \le 10$
  • Inputs 7-9: $N \le 80$
  • Inputs 10-17: $N \le 300$
  • Inputs 18-20: $N \le 500$

Problem credits: Alex Liang

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