Suppose we fix one circular permutation of n elements. Then I want to count the number of ways to rearrange them so that all elements has different element at its right side from the fixed circular permutation.
For instance, if we consider circular permutation of [4], and fix (1,2,3,4), the only way to rearrange this in. such way is (1,4,3,2). (1,3,4,2) is not good as 2's right is 1.
I considered this as mapping from [n] to [n] given by $f(n) $ is the element placed at right side of n. Then count mapping that no elements are mapped to same elements as a fixed mapping. I can not generalize this to any natural number, only countable when n is small...
You do not need to think this as mapping since it is just my guess. Any help is appreciated.