CIIM 2015 Problem 5

There are $n$ people seated on a circular table that have seats numerated from 1 to $n$ clockwise. Let $k$ be a fix integer with $2 \leq k \leq n$ . The people can change their seats. There are two types of moves permitted:

1. Each person moves to the next seat clockwise.
2. Only the ones in seats 1 and $k$ exchange their seats.

Determine, in function of $n$ and $k$ , the number of possible configurations of people in the table that can be attain by using a sequence of permitted moves.