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).