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F - Two Exams

Score : $500$ points

Problem Statement

In the Kingdom of Takahashi, $N$ citizens numbered $1$ to $N$ took an examination of competitive programming.
There were two tests, and Citizen $i$ ranked $P_i$-th in the first test and $Q_i$-th in the second test.
There were no ties in either test. That is, each of the sequences $P$ and $Q$ is a permutation of $(1, 2, ..., N)$.

Iroha, the president of the kingdom, is going to select $K$ citizens for the national team at the coming world championship of competitive programming.
The members of the national team must be selected so that the following is satisfied.

• There should not be a pair of citizens $(x, y)$ where Citizen $x$ is selected and Citizen $y$ is not selected such that $P_x > P_y$ and $Q_x > Q_y$.
  • In other words, if Citizen $y$ got a rank smaller than that of Citizen $x$ in both tests, it is not allowed to select Citizen $x$ while not selecting Citizen $y$.

To begin with, Iroha wants to know the number of ways to select citizens for the national team that satisfy the condition above. Find it to help her.
Since this number can be enormous, print it modulo $998244353$.

Constraints

• All values in input are integers.
• $1 \le N \le 300$
• $1 \le K \le N$
• Each of $P$ and $Q$ is a permutation of $(1,2,...,N)$.

Input

Input is given from Standard Input in the following format:

$N$ $K$

$P_1$ $P_2$ $\dots$ $P_N$

$Q_1$ $Q_2$ $\dots$ $Q_N$

Output

Print the answer as an integer.

4 2
2 4 3 1
2 1 4 3

3

• It is fine to select Citizen $1$ and Citizen $2$ for the team.
• If Citizen $1$ and Citizen $3$ are selected, Citizen $4$ ranked higher than Citizen $3$ did in both tests, so the pair $(x,y)=(3,4)$ would violate the condition in the Problem Statement.
• It is fine to select Citizen $1$ and Citizen $4$.
• If Citizen $2$ and Citizen $3$ are selected, the pair $(x,y)=(3,1)$ would violate the condition.
• It is fine to select Citizen $2$ and Citizen $4$.
• If Citizen $3$ and Citizen $4$ are selected, the pair $(x,y)=(3,1)$ would violate the condition.

The final answer is $3$.

33 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

168558757

All $\binom{33}{16} = 1166803110$ ways of selecting $16$ from the $33$ citizens satisfy the requirement.
Therefore, we should print $1166803110$ modulo $998244353$, that is, $168558757$.

15 7
4 9 7 5 6 13 2 11 3 1 12 14 15 10 8
4 14 9 12 7 15 1 2 8 11 3 5 13 6 10

23