P15658 [ICPC 2025 Jakarta R] Count DFS Graph

You are currently researching a graph traversal algorithm called the Depth First Search (DFS). Starting with an empty list $\texttt{A}$, the following pseudocode will fill the list $\texttt{A}$ with the visitation order of a DFS algorithm. ``` DFS(v): append v to A for each u neighbour of v in ascending node number: if u is not in A: DFS(u) ``` After running $\texttt{DFS(1)}$ from the pseudocode above, you now have a list $A$ containing a permutation of integers from $1$ to $N$. You now wonder how many different simple undirected graphs with $N$ nodes there are that yield the list $A$ that you have. Count the number, modulo $998\,244\,353$. A graph is simple when there are no self-loops and there is at most one edge connecting each pair of nodes. Two graphs are considered different if there is an edge connecting a pair of nodes in one graph but not the other.

The first line contains an integer $N$ ($2 \le N \le 300$). The second line contains a permutation of the first $N$ positive integers, representing the list $A$. The first element of $A$ is guaranteed to be $1$.

A single integer representing the number of different graphs, whose DFS order gives you the list $A$, modulo $998\,244\,353$.

$\textit{Explanation of Sample 1:}$ The following illustrates all the graphs with the given DFS order. :::align{center}  :::