G - Connectivity 2

Score : $600$ points

Problem Statement

Given is a simple undirected graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered $1,2,\dots,N$, the edges are numbered $1,2,\dots,M$, and Edge $i$ connects Vertex $a_i$ and Vertex $b_i$.
Consider removing zero or more edges from $G$ to get a new graph $H$. There are $2^M$ graphs that we can get as $H$. Among them, find the number of such graphs that Vertex $1$ and Vertex $k$ are directly or indirectly connected, for each integer $k$ such that $2 \leq k \leq N$.
Since the counts may be enormous, print them modulo $998244353$.

Constraints

• $2 \leq N \leq 17$
• $0 \leq M \leq \frac{N(N-1)}{2}$
• $1 \leq a_i \lt b_i \leq N$
• $(a_i, b_i) \neq (a_j, b_j)$ if $i \neq j$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$a_1$ $b_1$

$\vdots$

$a_M$ $b_M$

Output

Print $N-1$ lines. The $i$-th line should contain the answer for $k = i + 1$.

3 2
1 2
2 3

2
1

We can get the following graphs as $H$.

• The graph with no edges. Vertex $1$ is disconnected from any other vertex.
• The graph with only the edge connecting Vertex $1$ and $2$. Vertex $2$ is reachable from Vertex $1$.
• The graph with only the edge connecting Vertex $2$ and $3$. Vertex $1$ is disconnected from any other vertex.
• The graph with both edges. Vertex $2$ and $3$ are reachable from Vertex $1$.

5 6
1 2
1 4
1 5
2 3
2 5
3 4

43
31
37
41

2 0

0