A11781. 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 $10^9 + 7$ .

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 $i+1$ .

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

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 $(1,1,1)$ , $(2,1,1)$ , $(2,2,1)$ , $(2,2,2)$ , $(3,1,1)$ , $(3,2,1)$ , $(3,2,2)$ , $(3,3,1)$ , $(3,3,2)$ , $(3,3,3)$ .