102095: [AtCoder]ABC209 F - Deforestation

Problem Statement

There are $N$ trees standing in a row from left to right. The $i$-th tree $(1 \leq i \leq N)$ from the left, Tree $i$, has the height of $H_i$.

You will now cut down all these $N$ trees in some order you like. Formally, you will choose a permutation $P$ of $(1, 2, \ldots, N)$ and do the operation below for each $i=1, 2, 3, ..., N$ in this order.

• Cut down Tree $P_i$, that is, set $H_{P_i}$ to $0$, at a cost of $H_{P_i-1}+H_{P_i}+H_{P_i+1}$.

Here, we assume $H_0=0,H_{N+1}=0$.

In other words, the cost of cutting down a tree is the total height of the tree and the neighboring trees just before doing so.

Find the number of permutations $P$ that minimize the total cost of cutting down the trees. Since the count may be enormous, print it modulo $(10^9+7)$.

Constraints

• $1 \leq N \leq 4000$
• $1 \leq H_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$
$H_1$ $H_2$ $\ldots$ $H_N$

Output

Print the number of permutations $P$, modulo $(10^9+7)$, that minimize the total cost of cutting down the trees.

Sample Input 1

3
4 2 4

Sample Output 1

2

There are two permutations $P$ that minimize the total cost: $(1,3,2)$ and $(3,1,2)$.

Below, we will show the process of cutting down the trees for $P=(1,3,2)$, for example.

• First, Tree $1$ is cut down at a cost of $H_0+H_1+H_2=6$.
• Next, Tree $3$ is cut down at a cost of $H_2+H_3+H_4=6$.
• Finally, Tree $2$ is cut down at a cost of $H_1+H_2+H_3=2$.

The total cost incurred is $14$.

Sample Input 2

3
100 100 100

Sample Output 2

6

Sample Input 3

15
804289384 846930887 681692778 714636916 957747794 424238336 719885387 649760493 596516650 189641422 25202363 350490028 783368691 102520060 44897764

Sample Output 3

54537651

Be sure to print the count modulo $(10^9+7)$.

问题描述

从左到右有$N$棵树。从左数第$i$棵，即树$i$，高度为$H_i$。

现在你要按照你喜欢的顺序砍倒所有$N$棵树。形式上，你需要选择一个$(1, 2, \ldots, N)$的排列$P$，并按照以下顺序进行操作：

• 砍倒树$P_i$，即令$H_{P_i}$为$0$，成本为$H_{P_i-1}+H_{P_i}+H_{P_i+1}$。

这里，我们假设$H_0=0,H_{N+1}=0$。

换句话说，砍倒一棵树的成本是在砍倒前该树及其相邻树的总高度。

找出能最小化砍倒所有树的总成本的排列$P$的数量。由于计数可能非常大，所以结果对$(10^9+7)$取模。

限制条件

• $1 \leq N \leq 4000$
• $1 \leq H_i \leq 10^9$
• 输入中的所有值都是整数。

输入

输入从标准输入中给出，格式如下：

$N$
$H_1$ $H_2$ $\ldots$ $H_N$

输出

打印出能最小化砍倒所有树的总成本的排列$P$的数量，对$(10^9+7)$取模。

样例输入1

3
4 2 4

样例输出1

2

有2种排列$P$可以最小化总成本：$(1,3,2)$和$(3,1,2)$。

例如，以下是按照$P=(1,3,2)$砍倒树的过程：

• 首先，树$1$被砍倒，成本为$H_0+H_1+H_2=6$。
• 接着，树$3$被砍倒，成本为$H_2+H_3+H_4=6$。
• 最后，树$2$被砍倒，成本为$H_1+H_2+H_3=2$。

总成本为$14$。

样例输入2

3
100 100 100

样例输出2

6

样例输入3

15
804289384 846930887 681692778 714636916 957747794 424238336 719885387 649760493 596516650 189641422 25202363 350490028 783368691 102520060 44897764

样例输出3

54537651

请确保对$(10^9+7)$取模输出计数。