A15076 | Decinc Dividing

Let's call an array $a$ of $m$ integers $a_1, a_2, \ldots, a_m$ Decinc if $a$ can be made increasing by removing a decreasing subsequence (possibly empty) from it.

- For example, if $a = [3, 2, 4, 1, 5]$ , we can remove the decreasing subsequence $[a_1, a_4]$ from $a$ and obtain $a = [2, 4, 5]$ , which is increasing.

You are given a permutation $p$ of numbers from $1$ to $n$ . Find the number of pairs of integers $(l, r)$ with $1 \le l \le r \le n$ such that $p[l \ldots r]$ (the subarray of $p$ from $l$ to $r$ ) is a Decinc array.

The first line contains a single integer $n$ ( $1 \le n \le 2 \cdot 10^5$ ) — the size of $p$ .

The second line contains $n$ integers $p_1, p_2, \ldots, p_n$ ( $1 \le p_i \le n$ , all $p_i$ are distinct) — elements of the permutation.

Output the number of pairs of integers $(l, r)$ such that $p[l \ldots r]$ (the subarray of $p$ from $l$ to $r$ ) is a Decinc array. $(1 \le l \le r \le n)$