Consider the set, $S$, of $n$-tuples defined inductively as follows:
- $(1, 2, \ldots, n) \in S$
- if $(x_1, x_2, \ldots, x_i, x_{i+1}, \ldots, x_n) \in S$, then $(x_{i+1}, \ldots, x_{n}, x_1, x_2, \ldots, x_{i}) \in S$
What is the name of these types of $n$-tuples?
Note, the $n$-tuple $(2, 4, 1, 3)$ demonstrates that $S$ is a strict subset of all permutations.