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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.

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    @joriki -- Thank you, you're right. This is indeed a cyclic permutation. I think I am confused. I meant to ask about a different set, I tried to formalize it, but I think I lost something along the way.2011-05-28

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These are cyclic permutations of the tuple $(1, 2, \ldots, n)$.

Thanks, @joriki.