Removing Gacha

AtCoder

IDarc150_d

Time2000ms

Memory256MB

Difficulty—

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: $N$ $p_2$ $p_3$ $\dots$ $p_{N}$ [samples]

Samples

Input #1

4
1 1 3

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 $\displaystyle \frac{35}{6}$.

Input #2

15
1 2 1 1 4 5 3 3 5 10 3 6 3 13

Output #2

515759610

API Response (JSON)

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    "name": "Removing Gacha",
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      "content": "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. Initi",
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    {
      "statement_type": "Markdown",
      "content": "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 $p_i$.\nEach vertex has a color: black or white. Initi...",
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