Shuffle

题意翻译

给定一个长度为 $n$ 的 $01$ 序列 $a$ ，你需要求出对 $a$ 进行**最多一次**以下操作后，可能的 $a$ 的数量，对 $998244353$ 取模： * 选定 $a$ 的一段连续的子段 $[l,r],1\le l\le r\le n$ ，满足子段内有恰好 $k$ 个 $1$ （即 $\sum_{i=l}^ra_i=k$ ）。 * 将 $[l,r]$ 内的元素任意排列。

题目描述

You are given a binary string (i. e. a string consisting of characters 0 and/or 1) $ s $ of length $ n $ . You can perform the following operation with the string $ s $ at most once: choose a substring (a contiguous subsequence) of $ s $ having exactly $ k $ characters 1 in it, and shuffle it (reorder the characters in the substring as you wish). Calculate the number of different strings which can be obtained from $ s $ by performing this operation at most once.

输入输出格式

输入格式

The first line contains two integers $ n $ and $ k $ ( $ 2 \le n \le 5000 $ ; $ 0 \le k \le n $ ). The second line contains the string $ s $ of length $ n $ , consisting of characters 0 and/or 1.

输出格式

Print one integer — the number of different strings which can be obtained from $ s $ by performing the described operation at most once. Since the answer can be large, output it modulo $ 998244353 $ .

输入输出样例

输入样例 #1

7 2
1100110

输出样例 #1

16

输入样例 #2

5 0
10010

输出样例 #2

1

输入样例 #3

8 1
10001000

输出样例 #3

10

输入样例 #4

10 8
0010011000

输出样例 #4

1

说明

Some strings you can obtain in the first example: - to obtain 0110110, you can take the substring from the $ 1 $ -st character to the $ 4 $ -th character, which is 1100, and reorder its characters to get 0110; - to obtain 1111000, you can take the substring from the $ 3 $ -rd character to the $ 7 $ -th character, which is 00110, and reorder its characters to get 11000; - to obtain 1100101, you can take the substring from the $ 5 $ -th character to the $ 7 $ -th character, which is 110, and reorder its characters to get 101. In the second example, $ k = 0 $ so you can only choose the substrings consisting only of 0 characters. Reordering them doesn't change the string at all, so the only string you can obtain is 10010.