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F - |LIS| = 3

Score : $500$ points

Problem Statement

Find the number of sequences that satisfy all of the conditions below, modulo $998244353$.

• The length is $N$.
• Each of the elements is an integer between $1$ and $M$ (inclusive).
• Its longest increasing subsequence has the length of exactly $3$.

Notes

A subsequence of a sequence is the result of removing zero or more elements from it and concatenating the remaining elements without changing the order. For example, $(10,30)$ is a subsequence of $(10,20,30)$, while $(20,10)$ is not a subsequence of $(10,20,30)$.

A longest increasing subsequence of a sequence is its strictly increasing subsequence with the greatest length.

Constraints

• $3 \leq N \leq 1000$
• $3 \leq M \leq 10$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

Output

Print the answer.

4 5

135

One sequence that satisfies the conditions is $(3,4,1,5)$.
On the other hand, $(4,4,1,5)$ do not, because its longest increasing subsequence has the length of $2$.

3 4

4

111 3

144980434

Find the count modulo $998244353$.