Two of the classes freshmen at the University of Texas often take are linear algebra and discrete math. The former teaches vectors, matrices, and operations on them, while the latter covers a variety of topics including combinatorics and graph theory.
John is a freshman at UT, and it is finals season. His professors gave study guides for their final exams, and one of the bullet points in the linear algebra study guide is the following:
Given a matrix, you should be able to compute its determinant.
Here’s a bullet point in the discrete math study guide:
Given a graph, you should be able to write down its adjacency matrix.
In the course of his studying, John realized that these problems can be combined naturally: given a graph, compute the determinant of its adjacency matrix. After solving this problem a few times, John began to grow bored, and he wondered if he could find some information about the answer to this question for a whole class of graphs. He wrote the following question on a blank page in his notebook before going to sleep:
Given a positive integer $N$, consider a uniform random labeled tree on $N$ vertices. What is the expected value of the determinant of its adjacency matrix?
Now John is asleep and the problem is haunting his dreams. Can you help him solve it?
It can be shown that the answer is a rational number that can be written in the form $p/q$, with $q \not\equiv 0 \pmod{998\, 244\, 353}$. You should output your answer as an integer in $[0, 998\, 244\, 353)$ equivalent to $pq^{-1}$ modulo $998\, 244\, 353$.
A tree on $N$ vertices is a connected undirected graph with $N$ vertices and $N - 1$ edges. A labeled tree is one whose vertices are labeled with the integers $1, \dots , N$. Two labeled trees are distinct if they do not have the same set of edges, where an edge is defined as an unordered pair of integers in $\{ 1, \dots , N\} $. Note that two distinct labeled trees may have the same structure, but different vertex labelings.
The input consists of a single line containing a single integer, $N$ ($1 \leq N \leq 10^6$), the number of vertices in the uniform random labeled tree.
Output a single integer in $[0, 998\, 244\, 353)$, the answer to the problem as a rational number modulo $998\, 244\, 353$.
Sample Input 1 | Sample Output 1 |
---|---|
1 |
0 |
Sample Input 2 | Sample Output 2 |
---|---|
2 |
998244352 |