Counting Heaps

You are given a rooted tree with $n$ vertices. Assign the numbers $1$ through $n$ to the vertices, using each number exactly once, so that every vertex receives a number smaller than its parent's number. In other words, the root holds the largest number and the numbers decrease from parents to children, forming a max-heap.

Count the number of such labelings. Since this number can be very large, print it modulo $m$.

The first line contains the number of test cases $t$. ($1 \le t \le 250$)

The first line of each test case contains the number of vertices $n$ and the modulus $m$. ($1 \le n \le 500{,}000$, $2 \le m \le 10^{9}$)

The $i$-th of the next $n-1$ lines contains $p_{i+1}$, the parent of vertex $i+1$. ($1 \le p_{i+1} \le i$) Vertex 1 is always the root. The total input size does not exceed 50MB.

For each test case, print the number of valid labelings modulo $m$, one per line.

For example, in the last case of the sample ($n=5$, where vertices $2,3,4,5$ have parents $1,1,3,3$), there are exactly $8$ valid labelings.

Note that $m$ is not necessarily prime, so you cannot directly divide (use a modular inverse) under the modulus.