codeforces#P1151E. Number of Components
Number of Components
Description
The Kingdom of Kremland is a tree (a connected undirected graph without cycles) consisting of $n$ vertices. Each vertex $i$ has its own value $a_i$. All vertices are connected in series by edges. Formally, for every $1 \leq i < n$ there is an edge between the vertices of $i$ and $i+1$.
Denote the function $f(l, r)$, which takes two integers $l$ and $r$ ($l \leq r$):
- We leave in the tree only vertices whose values range from $l$ to $r$.
- The value of the function will be the number of connected components in the new graph.
Your task is to calculate the following sum: $$\sum_{l=1}^{n} \sum_{r=l}^{n} f(l, r) $$
The first line contains a single integer $n$ ($1 \leq n \leq 10^5$) — the number of vertices in the tree.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq n$) — the values of the vertices.
Print one number — the answer to the problem.
Input
The first line contains a single integer $n$ ($1 \leq n \leq 10^5$) — the number of vertices in the tree.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq n$) — the values of the vertices.
Output
Print one number — the answer to the problem.
Samples
3
2 1 3
7
4
2 1 1 3
11
10
1 5 2 5 5 3 10 6 5 1
104
Note
In the first example, the function values will be as follows:
- $f(1, 1)=1$ (there is only a vertex with the number $2$, which forms one component)
- $f(1, 2)=1$ (there are vertices $1$ and $2$ that form one component)
- $f(1, 3)=1$ (all vertices remain, one component is obtained)
- $f(2, 2)=1$ (only vertex number $1$)
- $f(2, 3)=2$ (there are vertices $1$ and $3$ that form two components)
- $f(3, 3)=1$ (only vertex $3$)
In the second example, the function values will be as follows:
- $f(1, 1)=1$
- $f(1, 2)=1$
- $f(1, 3)=1$
- $f(1, 4)=1$
- $f(2, 2)=1$
- $f(2, 3)=2$
- $f(2, 4)=2$
- $f(3, 3)=1$
- $f(3, 4)=1$
- $f(4, 4)=0$ (there is no vertex left, so the number of components is $0$)