Combinatorics Through Guided Discovery, 2004a

6.2. Groups Acting on Sets 117
1 2
R R
B B
4 3
Figure 6.2.2: The colored square with coloring {(1, R), (2, R), (3, B), (4, B)}
This gives us an explicity list of which colors are assigned to which vertex.4
Then if we rotate the square through 90 degrees, we see that the set of ordered
pairs becomes {
(ρ(1), R), (ρ(2), R), (ρ(3), B), (ρ(4), B)
}
(6.2)
which is the same as
Or, in a more natural order,
{(2, R), (3, R), (4, B), (1, B)} .
{(1, B), (2, R), (3, R), (4, B)} . (6.3)
The reordering we did in (6.3) suggests yet another simplification of notation. So
long as we know we that the first elements of our pairs are labeled by the members
of [n] for some integer n and we are listing our pairs in increasing order by the first
component, we can denote the coloring
{(1, B), (2, R), (3, R), (4, B)}
by BRRB. In the case where we have numbered the elements of the set S we are
coloring, we will call this list of colors of the elements of S in order the standard
notation for the coloring. We will call the ordering used in (6.3)the standard
ordering of the coloring.
Thus we have three natural ways to represent a coloring of a set: as a function,
as a set of ordered pairs, and as a list. Different representations are useful for
different things. For example, the representation by ordered pairs will provide a
natural way to define the action of a group on colorings of a set. Given a coloring
as a function f , we denote the set of ordered pairs
{ }
(x, f (x)) | x ∈ S ,
suggestively as (S, f ) for short. We use f (1) f (2) ··· f (n) to stand for a particular
coloring (S, f ) in the standard notation.
Problem 284. Suppose now that instead of coloring the vertices of a square,
we color its edges. We will use the shorthand 12, 23, 34, and 41 to stand for
the edges of the square between vertex 1 and vertex 2, vertex 2 and vertex
4The reader who has studied Appendix A will recognize that this set of ordered pairs is the relation
of the function f , but we won’t need to make any specific references to the idea of a relation in what
follows.