Score : $500$ points
There are $N$ cards numbered $1,\ldots,N$. Card $i$ has $P_i$ written on the front and $Q_i$ written on the back.
Here, $P=(P_1,\ldots,P_N)$ and $Q=(Q_1,\ldots,Q_N)$ are permutations of $(1, 2, \dots, N)$.
How many ways are there to choose some of the $N$ cards such that the following condition is satisfied? Find the count modulo $998244353$.
Condition: every number $1,2,\ldots,N$ is written on at least one of the chosen cards.
Input is given from Standard Input in the following format:
$N$ $P_1$ $P_2$ $\ldots$ $P_N$ $Q_1$ $Q_2$ $\ldots$ $Q_N$
Print the answer.
3 1 2 3 2 1 3
3
For example, if you choose Cards $1$ and $3$, then $1$ is written on the front of Card $1$, $2$ on the back of Card $1$, and $3$ on the front of Card $3$, so this combination satisfies the condition.
There are $3$ ways to choose cards satisfying the condition: $\{1,3\},\{2,3\},\{1,2,3\}$.
5 2 3 5 4 1 4 2 1 3 5
12
8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
1