Score : $500$ points

问题描述

已知一个包含 $N$ 个整数的序列：$A = (A_1, A_2, \dots, A_N)$。

找出满足以下条件的（不一定连续的）子序列 $A'=(A'_1,A'_2,\ldots,A'_k)$ 的数量，其中子序列长度至少为 $2$：

• $A'_1 \leq A'_k$。

由于计数可能非常庞大，计算结果请模 $998244353$ 后输出。

这里需要注意的是，即使两个子序列在顺序上相同，但如果它们源自不同的索引集合，则被视为不同的子序列。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Given is a sequence of $N$ integers: $A = (A_1, A_2, \dots, A_N)$.

Find the number of (not necessarily contiguous) subsequences $A'=(A'_1,A'_2,\ldots,A'_k)$ of length at least $2$ that satisfy the following condition:

• $A'_1 \leq A'_k$.

Since the count can be enormous, print it modulo $998244353$.

Here, two subsequences are distinguished when they originate from different sets of indices, even if they are the same as sequences.

Constraints

• $2 \leq N \leq 3 \times 10^5$
• $1 \leq A_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print the number of (not necessarily contiguous) subsequences $A'=(A'_1,A'_2,\ldots,A'_k)$ of length at least $2$ that satisfy the condition in Problem Statement.

Sample Input 1

3
1 2 1

Sample Output 1

3

$A=(1,2,1)$ has four (not necessarily contiguous) subsequences of length at least $2$: $(1,2)$, $(1,1)$, $(2,1)$, $(1,2,1)$.

Three of them, $(1,2)$, $(1,1)$, $(1,2,1)$, satisfy the condition in Problem Statement.

Sample Input 2

3
1 2 2

Sample Output 2

4

Note that two subsequences are distinguished when they originate from different sets of indices, even if they are the same as sequences.

In this Sample, there are four subsequences, $(1,2)$, $(1,2)$, $(2,2)$, $(1,2,2)$, that satisfy the condition.

Sample Input 3

3
3 2 1

Sample Output 3

0

There may be no subsequence that satisfies the condition.

Sample Input 4

10
198495780 28463047 859606611 212983738 946249513 789612890 782044670 700201033 367981604 302538501

Sample Output 4

830

update @ 2024/3/10 09:45:19