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Contest: Task: Related: TaskF TaskH

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?
We start at the root and repeat the following procedure to list the vertices of the tree.
  • 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.
    • The list of vertices in the order they are recorded here is the preorder traversal of the tree.

    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 $998244353$.

    Sample Input 1

    4
1 2 4 3

    Sample Output 1

    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 $1,2,3,4$.

    Sample Input 2

    8
1 2 3 5 6 7 8 4

    Sample Output 2

    202