I - Red Black Tree

Time limit: 14 s
Memory limit: 1024 MiB
Languages: C, C++, Java, Pascal, ... (details)

You are given a rooted tree with $n$ nodes. The nodes are numbered $1..n$. The root is node $1$, and $m$ of the nodes are colored red, the rest are black.

You would like to choose a subset of nodes such that there is no node in your subset which is an ancestor of any other node in your subset. For example, if $A$ is the parent of $B$ and $B$ is the parent of $C$, then you could have at most one of $A$, $B$ or $C$ in your subset. In addition, you would like exactly $k$ of your chosen nodes to be red.

If exactly $m$ of the nodes are red, then for all $k = 0..m$, figure out how many ways you can choose subsets with $k$ red nodes, and no node is an ancestor of any other node.

Input

Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each test case will begin with a line with two integers $n$ $(1 \leq n \leq 2 \times 10^5)$ and $m$ $(0 \leq m \leq \text{min}(103 , n))$, where $n$ is the number of nodes in the tree, and $m$ is the number of nodes which are red. The nodes are numbered $1..n$.

Each of the next $n - 1$ lines will contain a single integer $p$ $(1 \leq p \leq n)$, which is the number of the parent of this node. The nodes are listed in order, starting with node $2$, then node $3$, and so on. Node $1$ is skipped, since it is the root. It is guaranteed that the nodes form a single tree, with a single root at node $1$ and no cycles.

Each of the next $m$ lines will contain single integer $r$ $(1 \leq r \leq n)$. These are the numbers of the red nodes. No value of $r$ will be repeated.

Output

Output $m + 1$ lines, corresponding to the number of subsets satisfying the given criteria with a number of red nodes equal to $k = 0..m$, in that order. Output this number modulo $10^9 + 7$.

Sample test(s)

Input
4 1 1 1 1 3
Output
5 4
Input
4 4 1 1 1 1 2 3 4
Output
1 4 3 1 0
Input
14 4 1 2 1 2 3 4 5 5 13 8 10 4 4 8 3 12 13
Output
100 169 90 16 0