201142: [AtCoder]ARC114 C - Sequence Scores
Memory Limit：1024 MB
Time Limit：2 S
Judge Style：Text Compare
Creator：
Submit：0
Solved：0
Description
Score : $600$ points
Problem Statement
For a sequence $A$ of length $N$ consisting of integers between $1$ and $M$ (inclusive), let us define $f(A)$ as follows:
- We have a sequence $X$ of length $N$, where every element is initially $0$. $f(A)$ is the minimum number of operations required to make $X$ equal $A$ by repeating the following operation:
- Specify $1 \leq l \leq r \leq N$ and $1 \leq v \leq M$, then replace $X_i$ with $\max(X_i, v)$ for each $l \leq i \leq r$.
Find the sum, modulo $998244353$, of $f(A)$ over all $M^N$ sequences that can be $A$.
Constraints
- $1 \leq N, M \leq 5000$
- All values in input are integers.
Input
Input is given from Standard Input in the following format:
$N$ $M$
Output
Print the sum of $f(A)$ over all sequences $A$, modulo $998244353$.
Sample Input 1
2 3
Sample Output 1
15
The $3 ^ 2 = 9$ sequences and the value of $f$ for those are as follows:
- For $A = (1, 1)$, we can make $X$ equal it with one operation with $(l = 1, r = 2, v = 1)$, so $f(A) = 1$.
- For $A = (1, 2)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 1)$ and $(l = 2, r = 2, v = 2)$, so $f(A) = 2$.
- For $A = (1, 3)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 1)$ and $(l = 2, r = 2, v = 3)$, so $f(A) = 2$.
- For $A = (2, 1)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 1)$ and $(l = 1, r = 1, v = 2)$, so $f(A) = 2$.
- For $A = (2, 2)$, we can make $X$ equal it with one operation with $(l = 1, r = 2, v = 2)$, so $f(A) = 1$.
- For $A = (2, 3)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 2)$ , $(l = 2, r = 2, v = 3)$, so $f(A) = 2$.
- For $A = (3, 1)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 1)$ and $(l = 1, r = 1, v = 3)$ so $f(A) = 2$.
- For $A = (3, 2)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 2)$ and $(l = 1, r = 1, v = 3)$, so $f(A) = 2$.
- For $A = (3, 3)$, we can make $X$ equal it with one operation with $(l = 1, r = 2, v = 3)$, so $f(A) = 1$.
The sum of these values is $3 \times 1 + 6 \times 2 = 15$.
Sample Input 2
3 2
Sample Output 2
15
Sample Input 3
34 56
Sample Output 3
649717324