Problem Statement

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

A permutation $x = (x_1, x_2, \cdots, x_N)$ of $(1, 2, \cdots, N)$ is judged to be a good permutation or not by the following criteria.

• Consider the following operation on $x$.
• Choose two adjacent elements $u$ and $v$ in $x$ such that $u$ and $v$ are in an ancestor-descendant relationship in $T$. It does not matter which is the ancestor and which is the descendant. Then, swap $u$ and $v$.
• If it is possible to obtain a permutation that is lexicographically strictly smaller than the initial state by performing the above operation zero or more times, $x$ is not a good permutation. If it is impossible to obtain a permutation lexicographically smaller than the initial state by any such operations, $x$ is a good permutation.

You are given a positive integer $B$. For a permutation $x$, define $\operatorname{hash}(x) = \sum_{1 \leq i \leq N} B^{i-1} \times x_i$.

Find the sum of $\operatorname{hash}(x)$ over all good permutations $x$, modulo $998244353$.

What is lexicographical order on sequences? A sequence $S = (S_1, S_2, \ldots, S_{|S|})$ is said to be lexicographically smaller than a sequence $T = (T_1, T_2, \ldots, T_{|T|})$ if and only if 1. or 2. below holds. Here, $|S|$ and $|T|$ denote the lengths of $S$ and $T$, respectively. $|S| \lt |T|$ and $(S_1, S_2, \ldots, S_{|S|}) = (T_1, T_2, \ldots, T_{|S|})$. There exists an integer $1 \leq i \leq \min\{ |S|, |T| \}$ such that the following two statements hold. $(S_1, S_2, \ldots, S_{i-1}) = (T_1, T_2, \ldots, T_{i-1})$. $S_i$ is smaller than $T_i$ (as a number).

Constraints

• $2 \leq N \leq 100$
• $1 \leq B < 998244353$
• $1 \leq P_i < i$ ($2 \leq i \leq N$)
• All input values are integers.

Input

The input is given from Standard Input in the following format:

N B
P_2 P_3 \cdots P_N

Output

Print the answer.