1622D · Shuffle

Problem: 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.

Input Format: 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.

Output Format: 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 $998244353$.

Note: 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.