Score : $300$ points

问题描述

你将得到一个包含 $N$ 个整数的序列 $C$。找出满足所有以下条件的整数序列 $A$ 的数量。

• 对于 $(1 \leq i \leq N)$，有 $1 \leq A_i \leq C_i$
• 对于 $(1 \leq i < j \leq N)$，有 $A_i \neq A_j$

由于计数可能非常巨大，请以模 $(10^9+7)$ 的形式输出结果。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a sequence $C$ of $N$ integers. Find the number of sequences $A$ of $N$ integers satisfying all of the following conditions.

• $1 \leq A_i \leq C_i\, (1 \leq i \leq N)$
• $A_i \neq A_j\, (1 \leq i < j \leq N)$

Since the count may be enormous, print it modulo $(10^9+7)$.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq C_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$C_1$ $C_2$ $\ldots$ $C_N$

Output

Print the number of sequences $A$ of $N$ integers satisfying all of the following conditions, modulo $(10^9+7)$.

Sample Input 1

2
1 3

Sample Output 1

2

We have two sequences $A$ satisfying all of the conditions: $(1,2)$ and $(1,3)$.
On the other hand, $A=(1,1)$, for example, does not satisfy the second condition.

Sample Input 2

4
3 3 4 4

Sample Output 2

12

Sample Input 3

2
1 1

Sample Output 3

0

We have no sequences $A$ satisfying all of the conditions, so we should print $0$.

Sample Input 4

10
999999917 999999914 999999923 999999985 999999907 999999965 999999914 999999908 999999951 999999979

Sample Output 4

405924645

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

update @ 2024/3/10 09:22:43