Three Minimums

8000ms

1048576K

Description:

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 $$$998\,244\,353$$$.

Input:

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 ">".

Output:

Print a single integer: the number of good permutations satisfying the constraints described in the statement, modulo $$$998\,244\,353$$$.

Sample Input:

5 3
>>>

Sample Output:

Sample Input:

5 1
<

Sample Output:

Sample Input:

6 5
<<><>

Sample Output:

Sample Input:

10 5
><<><

Sample Output:

Sample Input:

1008 20
<><<>>><<<<<>>>>>>>>

Sample Output:

284142857

Note:

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 $$$s$$$: $$$[1, 2, 4, 3, 6, 5]$$$, $$$[1, 2, 5, 3, 6, 4]$$$, $$$[1, 2, 6, 3, 5, 4]$$$, $$$[1, 3, 4, 2, 6, 5]$$$, $$$[1, 3, 5, 2, 6, 4]$$$, $$$[1, 3, 6, 2, 5, 4]$$$, $$$[1, 4, 5, 2, 6, 3]$$$, $$$[1, 4, 6, 2, 5, 3]$$$, $$$[1, 5, 6, 2, 4, 3]$$$, $$$[2, 3, 4, 1, 6, 5]$$$, $$$[2, 3, 5, 1, 6, 4]$$$, $$$[2, 3, 6, 1, 5, 4]$$$, $$$[2, 4, 5, 1, 6, 3]$$$, $$$[2, 4, 6, 1, 5, 3]$$$, $$$[2, 5, 6, 1, 4, 3]$$$, $$$[3, 4, 5, 1, 6, 2]$$$, $$$[3, 4, 5, 2, 6, 1]$$$, $$$[3, 4, 6, 1, 5, 2]$$$, $$$[3, 4, 6, 2, 5, 1]$$$, $$$[3, 5, 6, 1, 4, 2]$$$, $$$[3, 5, 6, 2, 4, 1]$$$, $$$[4, 5, 6, 1, 3, 2]$$$, $$$[4, 5, 6, 2, 3, 1]$$$.

Informação

Provedor Codeforces

Código CF1654H