A16099 | Non-Intersecting Subpermutations

You are given two integers $n$ and $k$ .

For an array of length $n$ , let's define its cost as the maximum number of contiguous subarrays of this array that can be chosen so that:

- each element belongs to at most one subarray;
- the length of each subarray is exactly $k$ ;
- each subarray contains each integer from $1$ to $k$ exactly once.

For example, if $n = 10$ , $k = 3$ and the array is $[1, 2, 1, 3, 2, 3, 2, 3, 1, 3]$ , its cost is $2$ because, for example, we can choose the subarrays from the $2$ -nd element to the $4$ -th element and from the $7$ -th element to the $9$ -th element, and we can show that it's impossible to choose more than $2$ subarrays.

Calculate the sum of costs over all arrays of length $n$ consisting of integers from $1$ to $k$ , and print it modulo $998244353$ .

The only line of the input contains two integers $n$ and $k$ ( $2 \le k \le n \le 4000$ ).

Print one integer — the sum of costs of all arrays of length $n$ consisting of integers from $1$ to $k$ taken modulo $998244353$ .