Circular Permutation
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).
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