Chromatic Polynomials
--------- -----------

Given an undirected graph G (a set of vertices and
edges) let P(G,n) be the number of ways to color G with
n colors so no adjacent vertices have the same color.

If e is an edge of G between vertices x and y, then

    P(G,n) = number of ways to color G such that no 2
             adjacent vertices OTHER THAN x or y have
	     the same color

	   - number of ways to color G such that no 2
             adjacent vertices OTHER THAN x or y have
	     the same color, AND x and y do have the
	     same color

	   = P(G',n) - P(G'',n)

where G' is the graph made from G by deleting edge e,
and G'' is the graph made from G by merging x and y,
deleting e, and deleting any duplicates of edges in
the resulting graph.  That is, if in G e1 is an edge
from x to z and e2 is an edge from y to z then in G''
x and y become the same vertex so e1 and e2 are now the
same edge and one of these must be deleted to avoid
duplicate edges.

G' and G'' both have fewer edges than G.  By repeating
this process you can reduce the problem to computing
the number of ways of coloring a graph with no edges
with n colors.  This is just n**d, where d is the number
of vertices in the graph with no edges.

Therefore P(G,n) is a polynomial in n of degree |G|,
where |G| is the number of vertices in G.  You are asked
to compute P(G,n) for various G.


Input
-----

For each of several test cases, a specification of a
graph G as follows:

   A line containing the number V of vertices.
   1 <= V <= 10.

   V lines each containing V binary digits
     (`0's and `1's).

Vertices are identified by integers i, 1 <= i <= V.
Lines of digits are numbered 1, 2, 3, from the first
line to the last line.  Digits in a line are numbered
1, 2, 3, from left to right. 

For 1 <= i,j <= V, digit j of line i is `1' if vertex
i is adjacent to vertex j, and `0' otherwise.  Digit
j of line i equals digit i of line j, and digit i of
line i is `0' (a vertex is NOT adjacent to itself).

No lines contain any spaces.  The input terminates
with an end of file.

Output
------

For each case, a single line containing V+1 integers,
which are the coefficients of P(G,n) from high order
to low order.


Example Input
------- -----

3
000
000
000
3
011
101
110
5
01000
10100
01010
00101
00010
5
01001
10100
01010
00101
10010


Example Output
------- ------

1 0 0 0
1 -3 2 0
1 -4 6 -4 1 0
1 -5 10 -10 4 0


File:	   chromatic.txt
Author:	   Bob Walton <walton@deas.harvard.edu>
Date:	   Mon Oct 17 03:05:49 EDT 2005

The authors have placed this file in the public domain;
they make no warranty and accept no liability for this
file.

RCS Info (may not be true date or author):

    $Author: walton $
    $Date: 2005/10/17 07:06:04 $
    $RCSfile: chromatic.txt,v $
    $Revision: 1.3 $