A15999 | Multiple of Three Cycles

An array $a_1,\dots,a_n$ of length $n$ is initially all blank. There are $n$ updates where one entry of $a$ is updated to some number, such that $a$ becomes a permutation of $1,2,\dots,n$ after all the updates.

After each update, find the number of ways (modulo $998\,244\,353$ ) to fill in the remaining blank entries of $a$ so that $a$ becomes a permutation of $1,2,\dots,n$ and all cycle lengths in $a$ are multiples of $3$ .

A permutation of $1,2,\dots,n$ is an array of length $n$ consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. A cycle in a permutation $a$ is a sequence of pairwise distinct integers $(i_1,\dots,i_k)$ such that $i_2 = a_{i_1},i_3 = a_{i_2},\dots,i_k = a_{i_{k-1}},i_1 = a_{i_k}$ . The length of this cycle is the number $k$ , which is a multiple of $3$ if and only if $k \equiv 0 \pmod 3$ .

The first line contains a single integer $n$ ( $3 \le n \le 3 \cdot 10^5$ , $n \equiv 0 \pmod 3$ ).

The $i$ -th of the next $n$ lines contains two integers $x_i$ and $y_i$ , representing that the $i$ -th update changes $a_{x_i}$ to $y_i$ .

It is guaranteed that $x_1,\dots,x_n$ and $y_1,\dots,y_n$ are permutations of $1,2,\dots,n$ , i.e. $a$ becomes a permutation of $1,2,\dots,n$ after all the updates.

Output $n$ lines: the number of ways (modulo $998\,244\,353$ ) after the first $1,2,\dots,n$ updates.