Score : $700$ points

Problem Statement

Given is a directed tree with $N$ vertices numbered $1$ through $N$.

Vertex $1$ is the root of the tree. For each integer $i$ such that $2 \leq i \leq N$, there is a directed edge from Vertex $p_i$ to $i$.

Let $a$ be the permutation of $(1, \ldots, N)$, and $a_i$ be the $i$-th element of $a$.

Among the $N!$ sequences that can be $a$, find the number of sequences that satisfy the following condition, modulo $998244353$.

• Condition: For every integer $i$ such that $1 \leq i \leq N$, Vertex $i$ is unreachable from Vertex $a_i$ by traversing one or more edges.

Constraints

• All values in input are integers.
• $1 \leq N \leq 2000$
• $1 \leq p_i < i$

Input

Input is given from Standard Input in the following format:

$N$

$p_{2}$ $\cdots$ $p_N$

Output

Print the number of sequences $a$ among the permutations of $(1, \ldots, N)$ that satisfy the condition in Problem Statement, modulo $998244353$.

4
1 1 3

4

30
1 1 3 1 5 1 1 1 8 9 7 3 11 11 15 14 4 10 11 12 1 10 13 11 7 23 8 12 18

746746186

• Remember to print the count modulo $998244353$.