CF995F.Cowmpany Cowmpensation

Allen, having graduated from the MOO Institute of Techcowlogy (MIT), has started a startup! Allen is the president of his startup. He also hires $n-1$ other employees, each of which is assigned a direct superior. If $u$ is a superior of $v$ and $v$ is a superior of $w$ then also $u$ is a superior of $w$. Additionally, there are no $u$ and $v$ such that $u$ is the superior of $v$ and $v$ is the superior of $u$. Allen himself has no superior. Allen is employee number $1$, and the others are employee numbers $2$ through $n$.

Finally, Allen must assign salaries to each employee in the company including himself. Due to budget constraints, each employee's salary is an integer between $1$ and $D$. Additionally, no employee can make strictly more than his superior.

Help Allen find the number of ways to assign salaries. As this number may be large, output it modulo $10^9 + 7$.

艾伦从“哞”理工学院（MIT）毕业后，创办了一家初创公司！艾伦担任这家公司的总裁。他还雇佣了 $n-1$ 名其他员工，每位员工都被指派了一位直属上级。若 $u$ 是 $v$ 的上级，且 $v$ 是 $w$ 的上级，则 $u$ 也是 $w$ 的上级。此外，不存在员工 $u$ 和 $v$，使得 $u$ 是 $v$ 的上级，同时 $v$ 又是 $u$ 的上级。艾伦本人没有上级。艾伦是第 $1$ 号员工，其余员工编号为 $2$ 至 $n$。

最后，艾伦必须为公司中的每位员工（包括他自己）分配薪水。由于预算限制，每位员工的薪水必须是介于 $1$ 和 $D$ 之间的整数。此外，任何员工的薪水都不能严格高于其直属上级的薪水。

请帮助艾伦计算分配薪水的方案总数。由于该数目可能很大，请输出结果对 $10^9 + 7$ 取模后的值。

The first line of the input contains two integers $n$ and $D$ ($1 \le n \le 3000$, $1 \le D \le 10^9$).

The remaining $n-1$ lines each contain a single positive integer, where the $i$-th line contains the integer $p_i$ ($1 \le p_i \le i$). $p_i$ denotes the direct superior of employee $i+1$.

输入的第一行包含两个整数 $n$ 和 $D$（$1 \le n \le 3000$，$1 \le D \le 10^9$）。

接下来的 $n-1$ 行每行包含一个正整数，其中第 $i$ 行包含整数 $p_i$（$1 \le p_i \le i$）。$p_i$ 表示员工 $i+1$ 的直接上级。

Output a single integer: the number of ways to assign salaries modulo $10^9 + 7$.

输出一个整数：分配薪水的方案数对 $10^9 + 7$ 取模的结果。

In the first sample case, employee 2 and 3 report directly to Allen. The three salaries, in order, can be $(1,1,1)$, $(2,1,1)$, $(2,1,2)$, $(2,2,1)$ or $(2,2,2)$.

In the second sample case, employee 2 reports to Allen and employee 3 reports to employee 2. In order, the possible salaries are $(1,1,1)$, $(2,1,1)$, $(2,2,1)$, $(2,2,2)$, $(3,1,1)$, $(3,2,1)$, $(3,2,2)$, $(3,3,1)$, $(3,3,2)$, $(3,3,3)$.

在第一个样例中，员工 2 和 3 直接向艾伦汇报。按顺序，三种薪水可能为 $(1,1,1)$、$(2,1,1)$、$(2,1,2)$、$(2,2,1)$ 或 $(2,2,2)$。

在第二个样例中，员工 2 向艾伦汇报，员工 3 向员工 2 汇报。按顺序，可能的薪水为 $(1,1,1)$、$(2,1,1)$、$(2,2,1)$、$(2,2,2)$、$(3,1,1)$、$(3,2,1)$、$(3,2,2)$、$(3,3,1)$、$(3,3,2)$、$(3,3,3)$。