Non-Intersecting Subpermutations

2000ms

524288K

Description:

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 $$$998244353$$$.

Input:

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

Output:

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

Sample Input:

10 3

Sample Output:

Sample Input:

2 2

Sample Output:

Sample Input:

1337 42

Sample Output:

524933698

Informação

Provedor Codeforces

Código CF1861E