A circular permutation is an arrangement of 
distinct objects along a fixed (i.e.,
cannot be picked up out of the plane and turned over) circle, where 
is a positive integer.
The count is
 |
The number is 
instead of the usual factorial 
since all cyclic permutations
of objects are equivalent because the circle can be rotated.
Equivalently, a circular permutation specifies a cyclic
order on the objects.
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).
