Degree Sequence in DFS Order

AtCoder

IDagc056_f

Time2000ms

Memory256MB

Difficulty—

Given are integers $N$ and $M$. Find the number of integer sequences $a=(a_1,a_2,\cdots,a_N)$ that can be generated as follows, modulo $998244353$. * Provide an undirected connected graph $G$ with $N$ vertices and $M$ edges. Here, $G$ may not contain self-loops, but **may contain multi-edges**. * Perform DFS on $G$ and let $a_i$ be the degree of the $i$\-th vertex to be visited. More accurately, execute the pseudo-code below to get $a$. a = empty array dfs(v): visited\[v\]=True a.append(degree\[v\]) for u in g\[v\]: if not visited\[u\]: dfs(u) dfs(arbitrary root) Here, the variable $g$ represents the graph $G$ as an adjacency list, where $g[v]$ is the list of vertices adjacent to the vertex $v$ in **arbitrary order**. For example, when $N=4,M=5$, it is possible to generate $a=(2,4,1,3)$, by providing the graph $G$ as follows. ![image](https://img.atcoder.jp/agc056/3bfec17f881ae4cd27eccae94ebeae10.png) Here, the numbers written on vertices represent the order they are visited in the DFS. (The DFS starts at the vertex labeled $1$.) The orange arrows show the transitions in the DFS, and the numbers next to them represent the order the edges are traversed. ## Constraints * $2 \leq N \leq M \leq 10^6$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

2 2

Output #1

1

The only possible result is $a=(2,2)$. Note that $G$ may have multi-edges.

Input #2

3 4

Output #2

Input #3

10 20

Output #3

186225754

Input #4

100000 1000000

Output #4

191021899

API Response (JSON)

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