A14898 | Three Minimums
时间限制1s
内存限制256MB
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题目描述
Given a list of distinct values, we denote with first minimum, second minimum, and third minimum the three smallest values (in increasing order).
A permutation $p_1, p_2, \dots, p_n$ is good if the following statement holds for all pairs $(l,r)$ with $1\le l < l+2 \le r\le n$ .
- If $\{p_l, p_r\}$ are (not necessarily in this order) the first and second minimum of $p_l, p_{l+1}, \dots, p_r$ then the third minimum of $p_l, p_{l+1},\dots, p_r$ is either $p_{l+1}$ or $p_{r-1}$ .
You are given an integer $n$ and a string $s$ of length $m$ consisting of characters "<" and ">".
Count the number of good permutations $p_1, p_2,\dots, p_n$ such that, for all $1\le i\le m$ ,
- $p_i < p_{i+1}$ if $s_i =$ "<";
- $p_i > p_{i+1}$ if $s_i =$ ">".
As the result can be very large, you should print it modulo $998\,244\,353$ .
A permutation $p_1, p_2, \dots, p_n$ is good if the following statement holds for all pairs $(l,r)$ with $1\le l < l+2 \le r\le n$ .
- If $\{p_l, p_r\}$ are (not necessarily in this order) the first and second minimum of $p_l, p_{l+1}, \dots, p_r$ then the third minimum of $p_l, p_{l+1},\dots, p_r$ is either $p_{l+1}$ or $p_{r-1}$ .
You are given an integer $n$ and a string $s$ of length $m$ consisting of characters "<" and ">".
Count the number of good permutations $p_1, p_2,\dots, p_n$ such that, for all $1\le i\le m$ ,
- $p_i < p_{i+1}$ if $s_i =$ "<";
- $p_i > p_{i+1}$ if $s_i =$ ">".
As the result can be very large, you should print it modulo $998\,244\,353$ .
输入格式
The first line contains two integers $n$ and $m$ ( $2 \le n \le 2 \cdot 10^5$ , $1 \leq m \leq \min(100, n-1)$ ).
The second line contains a string $s$ of length $m$ , consisting of characters "<" and ">".
The second line contains a string $s$ of length $m$ , consisting of characters "<" and ">".
输出格式
Print a single integer: the number of good permutations satisfying the constraints described in the statement, modulo $998\,244\,353$ .
输入输出样例
输入 #1
5 3 >>>
输出 #1
5
输入 #2
5 1 <
输出 #2
56
输入 #3
6 5 <<><>
输出 #3
23
输入 #4
10 5 ><<><
输出 #4
83154
输入 #5
1008 20 <><<>>><<<<<>>>>>>>>
输出 #5
284142857
In the first test, there are $5$ good permutations satisfying the constraints given by the string $s$ : $[4, 3, 2, 1, 5]$ , $[5, 3, 2, 1, 4]$ , $[5, 4, 2, 1, 3]$ , $[5, 4, 3, 1, 2]$ , $[5, 4, 3, 2, 1]$ . Each of them
- is good;
- satisfies $p_1 > p_2$ ;
- satisfies $p_2 > p_3$ ;
- satisfies $p_3 > p_4$ .
In the second test, there are $60$ permutations such that $p_1 < p_2$ . Only $56$ of them are good: the permutations $[1, 4, 3, 5, 2]$ , $[1, 5, 3, 4, 2]$ , $[2, 4, 3, 5, 1]$ , $[2, 5, 3, 4, 1]$ are not good because the required condition doesn't hold for $(l, r)$ = $(1, 5)$ . For example, for the permutation $[2, 4, 3, 5, 1]$ ,
- the first minimum and the second minimum are $p_5$ and $p_1$ , respectively (so they are $\{p_l, p_r\}$ up to reordering);
- the third minimum is $p_3$ (neither $p_{l+1}$ nor $p_{r-1}$ ).
In the third test, there are $23$ good permutations satisfying the constraints given by the string $s$ : $[1, 2, 4, 3, 6, 5]$ , $[1, 2, 5, 3, 6, 4]$ , $[1, 2, 6, 3, 5, 4]$ , $[1, 3, 4, 2, 6, 5]$ , $[1, 3, 5, 2, 6, 4]$ , $[1, 3, 6, 2, 5, 4]$ , $[1, 4, 5, 2, 6, 3]$ , $[1, 4, 6, 2, 5, 3]$ , $[1, 5, 6, 2, 4, 3]$ , $[2, 3, 4, 1, 6, 5]$ , $[2, 3, 5, 1, 6, 4]$ , $[2, 3, 6, 1, 5, 4]$ , $[2, 4, 5, 1, 6, 3]$ , $[2, 4, 6, 1, 5, 3]$ , $[2, 5, 6, 1, 4, 3]$ , $[3, 4, 5, 1, 6, 2]$ , $[3, 4, 5, 2, 6, 1]$ , $[3, 4, 6, 1, 5, 2]$ , $[3, 4, 6, 2, 5, 1]$ , $[3, 5, 6, 1, 4, 2]$ , $[3, 5, 6, 2, 4, 1]$ , $[4, 5, 6, 1, 3, 2]$ , $[4, 5, 6, 2, 3, 1]$ .
- is good;
- satisfies $p_1 > p_2$ ;
- satisfies $p_2 > p_3$ ;
- satisfies $p_3 > p_4$ .
In the second test, there are $60$ permutations such that $p_1 < p_2$ . Only $56$ of them are good: the permutations $[1, 4, 3, 5, 2]$ , $[1, 5, 3, 4, 2]$ , $[2, 4, 3, 5, 1]$ , $[2, 5, 3, 4, 1]$ are not good because the required condition doesn't hold for $(l, r)$ = $(1, 5)$ . For example, for the permutation $[2, 4, 3, 5, 1]$ ,
- the first minimum and the second minimum are $p_5$ and $p_1$ , respectively (so they are $\{p_l, p_r\}$ up to reordering);
- the third minimum is $p_3$ (neither $p_{l+1}$ nor $p_{r-1}$ ).
In the third test, there are $23$ good permutations satisfying the constraints given by the string $s$ : $[1, 2, 4, 3, 6, 5]$ , $[1, 2, 5, 3, 6, 4]$ , $[1, 2, 6, 3, 5, 4]$ , $[1, 3, 4, 2, 6, 5]$ , $[1, 3, 5, 2, 6, 4]$ , $[1, 3, 6, 2, 5, 4]$ , $[1, 4, 5, 2, 6, 3]$ , $[1, 4, 6, 2, 5, 3]$ , $[1, 5, 6, 2, 4, 3]$ , $[2, 3, 4, 1, 6, 5]$ , $[2, 3, 5, 1, 6, 4]$ , $[2, 3, 6, 1, 5, 4]$ , $[2, 4, 5, 1, 6, 3]$ , $[2, 4, 6, 1, 5, 3]$ , $[2, 5, 6, 1, 4, 3]$ , $[3, 4, 5, 1, 6, 2]$ , $[3, 4, 5, 2, 6, 1]$ , $[3, 4, 6, 1, 5, 2]$ , $[3, 4, 6, 2, 5, 1]$ , $[3, 5, 6, 1, 4, 2]$ , $[3, 5, 6, 2, 4, 1]$ , $[4, 5, 6, 1, 3, 2]$ , $[4, 5, 6, 2, 3, 1]$ .
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评测结果：Accepted