The original problem is from Iberoamerican University Competition 2015, Problem 5. It's statement is:
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:
- Each person moves to the next seat clockwise.
- 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 attained by using a sequence of permitted moves.
My attempt was the following.
We can consider a move as acting by a permutation on the set of $n$ people, and the only permitted permutations are $\tau=\begin{pmatrix} 1 & k \\ k &1 \end{pmatrix}$ and $\sigma=\begin{pmatrix} 1 & 2 &\ldots &n \\ 2 & 3 & \ldots & 1 \end{pmatrix}$ and assume the original seating is the identity, then the set of different seatings is the subgroup of $S_n$ generated by $\sigma$ and $\tau$. One also sees that $\mathrm{ord}(\sigma)=n$ and $\mathrm{ord}(\tau)=2$, we would also like to get an expression between the $\sigma$ and $\tau$ like the one in the Dihedral group and then we could find the cardinality of $\langle \tau, \sigma \rangle$. Here's where I'm stuck! Can anyone continue this approach?