The number of ways to arrange 
distinct objects along a fixed (i.e.,
cannot be picked up out of the plane and turned over) circle
is
 |
The number is 
instead of the usual factorial 
since all cyclic permutations
of objects are equivalent because the circle can be rotated.
For example, of the 
permutations of three objects, the 
distinct circular permutations are 
and 
. Similarly, of the 
permutations of four objects, the 
distinct circular permutations are 
, 
, 
, 
, 
, and 
. Of these, there are only three free
permutations (i.e., inequivalent when flipping the circle is allowed): 
, 
, and 
. The number of free circular permutations of order
is 
for 
,
2, and
 |
for 
,
giving the sequence 1, 1, 1, 3, 12, 60, 360, 2520, ... (OEIS A001710).
