Score : $600$ points

Problem Statement

There is a rooted tree with $N$ vertices called Vertex $1$, $2$, $\ldots$, $N$, rooted at Vertex $1$.

We performed a depth-first search starting at the root and obtained a preorder traversal of the tree: $P_1, P_2, \ldots, P_N$. During the search, when the current vertex had multiple children, we chose the unvisited vertex with the smallest index.

What is a preorder traversal?

• If the current vertex $u$ is not recorded yet, record it.

• Then, if $u$ has an unvisited vertex, go to that vertex.

• Otherwise, terminate if $u$ is the root, and go to the parent of $u$ if it is not.

Find the number of rooted trees consistent with the preorder traversal, modulo $998244353$. Two rooted trees (with $N$ vertices and rooted at Vertex $1$) are considered different when there is a non-root vertex whose parent is different in the two trees.

Constraints

• $2 \leq N \leq 500$
• $1 \leq P_i\leq N$
• $P_1=1$
• All $P_i$ are distinct.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$P_1$ $P_2$ $\ldots$ $P_N$

Output

Print the number of rooted trees consistent with the preorder traversal, modulo $998244353$.

4
1 2 4 3

3

The rooted trees consistent with the preorder traversal are the three shown below, so the answer is $3$.

Note that the tree below does not count. This is because, among the children of Vertex $2$, we visit Vertex $3$ before visiting Vertex $4$, resulting in the preorder traversal $1,2,3,4$.

8
1 2 3 5 6 7 8 4

202