Min Max Subarrays

Note: The time limit for this problem is 3s, 1.5x the default.

You are given a length-$N$ integer array $a_1,a_2,\dots,a_N$ ($2\le N\le 10^6, 1\le a_i\le N$). Output the sum of the answers for the subproblem below over all $N(N+1)/2$ contiguous subarrays of $a$.

Given a nonempty list of integers, alternate the following operations (starting with the first operation) until the list has size exactly one.

1. Replace two consecutive integers in the list with their minimum.
2. Replace two consecutive integers in the list with their maximum.

Determine the maximum possible value of the final remaining integer.

For example,

[4, 10, 3] -> [4, 3] -> [4]
[3, 4, 10] -> [3, 10] -> [10]

In the first array, $(10, 3)$ is replaced by $\min(10, 3)=3$ and $(4, 3)$ is replaced by $\max(4, 3)=4$.

The first line contains $N$.

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

• Inputs 4-5: $N\le 100$
• Inputs 6-7: $N\le 5000$
• Inputs 8-9: $\max(a)\le 10$
• Inputs 10-13: No additional constraints.

The sum of the answer to the subproblem over all subarrays.

The answer for $[2]$ is $2$, the answer for $[1]$ is $1$, and the answer for $[2, 1]$ is $1$.

Thus, our output should be $2+1+1 = 4$.

Consider the subarray $[2, 4, 1, 3]$.

1. Applying the first operation on (1, 3), our new array is $[2, 4, 1]$.
2. Applying the second operation on (4, 1), our new array is $[2, 4]$.
3. Applying the third operation on (2, 4), our final number is $2$.

It can be proven that $2$ is the maximum possible value of the final number.