WOLFRAM|DEMONSTRATIONS PROJECT

Counting Necklaces

​
number of beads
allow reflections
view examples
​
The number of 12-bead necklaces under
C
12
using m colors is:
1
12
(4m+2
2
m
+2
3
m
+2
4
m
+
6
m
+
12
m
)
order of rotation
action on necklace
cycle type
necklaces fixed
elements of this order
1
R
0
12
z
1
12
m
1
2
R
180
6
z
2
6
m
1
3
R
120
4
z
3
4
m
2
4
R
90
3
z
4
3
m
2
6
R
60
2
z
6
2
m
2
12
R
30
1
z
12
1
m
4
​
A necklace is an
m
-coloring of the vertices of a regular
n
-gon. The vertices are interpreted as the beads of the necklace. This Demonstration shows the computations involved in using Burnside's lemma to count the number of
n
-bead necklaces for
n=3
through
n=35
. Several examples are provided to clarify the notation used and illustrate necklaces fixed by rotation or reflection.