Score : $600$ points

问题描述

我们有一个包含 $N$ 个顶点的有根树 $T$，顶点编号为 $1$ 到 $N$。顶点 $1$ 是树的根节点，且顶点 $i$ $(2 \leq i \leq N)$ 的父节点是顶点 $P_i$。

对于树 $T$ 的顶点集合 $V = \lbrace 1, 2,\dots, N\rbrace$ 中的一个非空子集 $S$，如果满足以下条件，则称其为一个好顶点集合：

• 对于 $S$ 中每一对不同的顶点 $(u, v)$，都满足：$u$ 不是 $v$ 的祖先。

对于每个 $K = 1, 2, \dots, N$，求恰好包含 $K$ 个顶点的好顶点集合的数量，结果对 $998244353$ 取模。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a rooted tree $T$ with $N$ vertices numbered $1$ to $N$. Vertex $1$ is the root, and the parent of vertex $i$ $(2 \leq i \leq N)$ is vertex $P_i$.

A non-empty subset $S$ of the vertex set $V = \lbrace 1, 2,\dots, N\rbrace$ of $T$ is said to be a good vertex set when it satisfies the following condition.

• For every pair of different vertices $(u, v)$ in $S$, the following holds: $u$ is not an ancestor of $v$.

For each $K = 1, 2, \dots, N$, find the number, modulo $998244353$, of good vertex sets with exactly $K$ vertices.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq P_i \lt i$
• 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 $N$ lines. The $i$-th line should contain the answer for $K = i$.

Sample Input 1

4
1 2 1

Sample Output 1

4
2
0
0

For each $1 \leq K \leq N$, the good vertex sets of size $K$ are listed below.

• $K=1$: $\lbrace 1 \rbrace, \lbrace 2 \rbrace, \lbrace 3 \rbrace, \lbrace 4 \rbrace$.
• $K=2$: $\lbrace 2, 4 \rbrace, \lbrace 3, 4 \rbrace$.
• $K=3,4$: There is none.

Sample Input 2

6
1 1 2 2 5

Sample Output 2

6
6
2
0
0
0

Sample Input 3

6
1 1 1 1 1

Sample Output 3

6
10
10
5
1
0

Sample Input 4

10
1 2 1 2 1 1 2 6 9

Sample Output 4

10
30
47
38
16
3
0
0
0
0

update @ 2024/3/10 11:22:31