201503: [AtCoder]ARC150 D - Removing Gacha
Description
Score : $700$ points
Problem Statement
We have a rooted tree with $N$ vertices numbered $1$ to $N$. Vertex $1$ is the root of the tree, and the parent of vertex $i\ (2\leq i)$ is vertex $p_i$.
Each vertex has a color: black or white. Initially, all vertices are white.
In this rooted tree, vertex $i$ is said to be good when all vertices on the simple path connecting vertices $1$ and $i$ (including vertices $1$ and $i$) are black, and bad otherwise.
Until all vertices are black, let us repeat the following operation: choose one vertex from the bad vertices uniformly at random, and repaint the chosen vertex black.
Find the expected value of the number of times we perform the operation, modulo $998244353$.
The definition of the expected value modulo $998244353$
It can be proved that the sought expected value is always a rational number. Additionally, when that value is represented as an irreducible fraction $\frac{P}{Q}$, it can also be proved that $Q \not \equiv 0 \pmod{998244353}$. Therefore, there is a unique integer $R$ such that $R \times Q \equiv P \pmod{998244353}$ and $0 \leq R < 998244353$. Find this $R$.
Constraints
- $2 \leq N \leq 2 \times 10^5$
- $1 \leq p_i < i$
- All values in the input are integers.
Input
The input is given from Standard Input in the following format:
$N$
$p_2$ $p_3$ $\dots$ $p_{N}$
Output
Print the answer.
Sample Input 1
4 1 1 3
Sample Output 1
831870300
Consider a case where the first three operations have chosen vertices $1$, $2$, and $4$ in this order. Then, vertices $1$ and $2$ are good, but vertex $4$ is still bad since vertex $3$, an ancestor of vertex $4$, is white. Thus, the fourth operation will choose vertex $3$ or $4$ uniformly at random.
The expected value of the number of times we perform the operation is $\displaystyle \frac{35}{6}$.
Sample Input 2
15 1 2 1 1 4 5 3 3 5 10 3 6 3 13
Sample Output 2
515759610